HAYS, Circuit Judge:

This is an appeal in an action brought under section 301(a) of the National Labor Relations Act, 29 U.S.C. § 185(a) (1970), by a union to compel arbitration of a dispute concerning the interpretation and validity of a clause in a collective bargaining agreement which permits the employer to subcontract work, but only to union shops. The district court ordered arbitration, but declared *222the entire subcontracting clause void as violating section 8(e) of the National Labor Relations Act, as amended, 29 U. S.C. § 158(e) (1970). The employer appeals, claiming that the district court should have voided only so much of the contract clause as limits subcontracting to union shops. We affirm.

The material facts are not disputed.

Sterling Optical Company [Sterling] operates a warehouse and optical laboratory in Brooklyn, New York, and a large number of optical stores in the greater New York metropolitan area. The Unit-, ed Optical Workers Union Local 408 [Local 408], an affiliate of the International Union of Electrical, Radio and Machine Workers, AFL-CIO, is the exclusive collective bargaining agent for Sterling employees. Since 1954, Sterling and Local 408 have been parties to a continuous series of collective bargaining agreements, the most recent of which was entered into as of April 30, 1973, and is currently effective.

The present dispute arises out of actions taken by Sterling in September 1973. On September 17, 1973, Sterling announced to the Local 408 business manager that it had decided to subcontract to a non-union shop certain laboratory work being performed at the Brooklyn plant by bargaining unit employees, a decision terminating the employment of 36 Local 408 members. Two days later the business manager sent a telegram to Sterling requesting a meeting, and on September 21 representatives of Sterling and Local 408 met and conferred. The union took the position that the proposed subcontracting violated various provisions in the 1973 collective bargaining agreement, the most pertinent of which is Article XXVIII, which reads as follows:

“It is agreed by and between the Parties that whereas the Employer send [sic] work out to be done in different establishments under different managements, therefore, upon the signing of this Agreement all said work shall be sent out to Union establishments.” 1

Sterling took the position that Article XXVIII did not have the effect of forbidding subcontracting because the union shop provision was a “hot cargo” clause offensive to section 8(e) of the National Labor Relations Act.

The parties failed to adjust their differences at the meeting. The Local 408 representative presented Sterling with a written demand for arbitration, as was its right under Article VIII of the 1973 agreement. Article VIII refers to arbitration all disputes that are not adjusted by representatives of Sterling and Local 408:

“Any and all grievances or disputes between the Employer and his employees which cannot be satisfactorily adjusted by a representative of the Employer and a duly authorized representative of the Union shall be referred to an Arbitrator to be selected by the American Arbitration Association whose decision in the matter shall be final and binding upon both Parties to this Agreement, even though one of the Parties shall fail to appear; and such award shall be enforceable in any Court of competent jurisdiction.”

Sterling resisted Local 408’s demand for arbitration because of the Article XXVIII issue, but it has not opposed arbitration as to issues raised by the contract clauses aside from Article XXVIII.

On September 25 Local 408 commenced this action to compel arbitration of the Article XXVIII issue. Sterling counterclaimed for a judgment declaring Article XXVIII void and unenforceable under section 8(e) insofar as it purports to limit Sterling’s right to subcontract. The parties cross-moved for summary judgment. On December 10 the district court entered its judgment ordering Sterling to submit the dispute to arbitration and declaring that the entirety of Article XXVIII “is violative of § 8(e) of the National Labor Relations Act as amended (29 U.S.C. § 158(e)) *223and is null and void and of no effect . . . In its accompanying memorandum it found that the subcontracting dispute is subject to arbitration under Article VIII of the 1973 agreement, but that the issue of the validity of Article XXVIII under section 8(e) is not subject to arbitration. In declaring Article XXVIII void in its entirety, the court rejected Sterling’s argument that only the language offensive to section 8(e) should be excised; that the text of Article XXVIII should be retained insofar as it permits subcontracting.

Sterling appealed, and the district court stayed arbitration pending appeal. Local 408 did not cross-appeal.

Sterling argues that the district court should have redacted only the language in Article XXVIII that is directly offensive to section 8(e), the language limiting subcontracting to union shops, rather than the entire Article. It contends that the text of Article XXVIII should be revised so that it reads as follows:

It is agreed by and between the parties that whereas the Employer sends work out to be done in different establishments under different managements.

It evidently hopes to use this ungainly fragment to convince the arbitrator that it had an unfettered right to subcontract.

The district court was correct in not rewriting Article XXVIII to express a meaning contrary to the parties’ original understanding. Article XXVIII embodies a compromise by its very terms. Local 408 gave Sterling the right to subcontract in exchange for Sterling’s promise to subcontract only to union shops. Sterling’s promise was valid when it was first given in 1954.2 The parties could not have foreseen that Article XXVIII would be rewritten by a court in a manner unilaterally beneficial to Sterling. To void the language won by Local 408 while effectuating the remainder of Article XXVIII would upset the balance struck at the bargaining table.

Sterling cites a number of eases decided in other circuits, of which Lewis v. Seanor Coal Co., 382 F.2d 437 (3d Cir. 1967), cert. denied, 390 U.S. 947, 88 S.Ct. 1035, 19 L.Ed.2d 1137 (1968), Meat & Highway Drivers Local 710 v. NLRB, 118 U.S.App.D.C. 287, 335 F.2d 709 (1964), and Truck Drivers Local 413 v. NLRB, 118 U.S.App.D.C. 149, 334 F.2d 539, cert. denied, 379 U.S. 916, 85 S.Ct. 264, 13 L.Ed.2d 186 (1964), are most closely applicable to these facts. It is claimed that these cases stand for the proposition that as a matter of law only the invalid portion of a partially invalid clause should be redacted. In reality, these cases stand for more limited propositions. Lewis v. Seanor Coal Co., supra, and Truck Drivers Local 413 v. NLRB, supra, hold that an entire collective bargaining contract need not fail merely because it includes a subcontracting clause violating section 8(e). 382 F.2d at 440, 334 F.2d at 549. In the present case the district court did not invalidate the entire agreement. In Meat & Highway Drivers Local 710 v. NLRB, supra, the court found that the text retained after excising objectionable hot cargo language benefitted both the union and employer, so that the retained text “would leave the total collective bargaining agreement in a state close to the actual agreement of the parties.” 335 F.2d at 717. In the present case, rewriting Article XXVIII along the lines suggested by Sterling would result in a fragment expressing a meaning nowhere near the original understanding of the parties.

II.

Were the issue before us we would have no difficulty holding that *224the district court erred in declaring Article XXVIII void since the validity of Article XXVIII under section 8(e) is an issue lying initially in the exclusive province of the arbitrator. However, we have no power to pass on this question because of the failure of the Union to appeal.

The district court’s holding “that the issue of whether Article XXVIII of the agreement contravenes section 8(e) of the Act is nonarbitrable” is based on an erroneous reading of our decision in Todd Shipyards Corp. v. Marine Workers Local 39, 344 F.2d 107 (2d Cir. 1965). In Todd Shipyards we held that an employer’s action under section 301 for a declaratory judgment that a hot cargo clause is unenforceable is not barred because of the NLRB’s concurrent jurisdiction. We were concerned with coordinating competing district court and NLRB jurisdiction and we struck the balance in favor of the forum chosen by the plaintiff. The present case involves coordinating section 301 jurisdiction with an arbitration clause which by its terms covers the dispute. These facts require consideration of policies distinct from those governing Todd Shipyards. In a section 301 action to compel arbitration the court must give full scope to an arbitration clause. Our national labor policy encourages the settlement of labor disputes through arbitration, see § 203(d) of the Labor Management Relations Act, 29 U.S.C. § 173(d) (1970), even to the extent, for example, of enforcing an arbitration clause despite the anti-injunction provisions of the Norris LaGuardia Act. See Boys Markets, Inc. v. Retail Clerk’s Local 770, 398 U.S. 235, 90 S.Ct. 1583, 26 L.Ed.2d 199 (1970). That policy limits the district court’s role in a section 301 action to compel arbitration to directing arbitration of disputes falling within the agreement. The scope of the court’s inquiry is accordingly limited: “It is confined to ascertaining whether the party seeking arbitration is making a claim which on its face is governed by the contract. Whether the moving party is right or wrong is a question of contract interpretation for the arbitrator.” United Steelworkers v. American Manufacturing Co., 363 U.S. 564, 568, 80 S.Ct. 1343, 1346, 4 L.Ed.2d 1403 (1960). In the present case it is clear that Local 408’s grievance comes within the arbitration clause and that its claim is governed by Article XXVIII. Whether the subcontracting clause violates section 8(e) is at least initially in the exclusive province of the arbitrator.

However, we cannot reverse the district court’s erroneous withdrawal of the Article XXVIII issue from arbitration in the procedural stance of this appeal. Local 408 did not file a cross-appeal, so we may not modify the district court order in its favor. Morley Construction Co. v. Maryland Casualty Co., 300 U.S. 185, 191, 57 S.Ct. 325, 81 L.Ed. 593 (1937). Of course it would have been proper for Local 408 to advance any matter in the record in support of the district court’s order, including arguments previously rejected by the district court, United States v. American Railway Express Co., 265 U.S. 425, 435, 44 S.Ct. 560, 68 L.Ed. 1087 (1924), 9 J. W. Moore, Federal Practice ¶ 204.11 [3] at 934-35 (2d ed. 1973), but it may not attempt to enlarge the relief granted by the lower court or lessen the rights of Sterling. Swarb v. Lennox, 405 U.S. 191, 201, 92 S.Ct. 767, 31 L.Ed.2d 138 (1972); United States v. American Railway Express Co., supra; Penfield Co. v. SEC, 330 U.S. 585, 594, 67 S.Ct. 918, 91 L.Ed. 1117 (1947); Schildhaus v. Moe, 319 F.2d 587 (2d Cir. 1963) (per curiam). It may well be that the cross-appeal rule is a rule of practice, see Langnes v. Green, 282 U.S. 531, 538, 51 S.Ct. 243, 75 L.Ed. 520 (1931), which we may dispense with in an appropriate case. Id., J. W. Moore, supra, at ¶ 204.11 [5], But the present case is not one calling for an exception to a rule so well-established and of such long usage.

The judgment of the district court is accordingly affirmed.

. The preamble to the 1973 agreement defines the term “Union” to mean Local 408.

. The present text of Article XXVIII appeared in 1954 in the first contract between these parties and has been used continuously to this date. Section 8(e) was not included in tiie National Labor Relations Act until 1959 when it was added by section 704(b) of the Labor Management Reporting and Disclosure Act of 1959, 73 Stat. 519 (1959).