B767-200 CREW HAD RWY INCURSION AT EBBR.

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WHILE TAXIING FOR TKOF AT BRUSSELS; BELGIUM; CLRNC WAS GIVEN BY GND CTL TO CROSS RWY 20 AT TXWY E4. THIS WAS A CHANGE FROM A PREVIOUS CLRNC TO TAXI ON OUTER 8 AND HOLD SHORT OF RWY 20 AT E5. THE FO WAS TALKING ON THE GND FREQ AND ASKED GND TO CONFIRM THAT WE WERE CLR TO CROSS! THE CAPT AND INTL RELIEF OFFICER WERE MONITORING THE FREQ AND UNDERSTOOD 'CLR TO CROSS RWY 20 AND CTL TWR ON THE OTHER SIDE.' WE CHKED FINAL APCH AND OBSERVED A TURBOPROP ON ABOUT A 3 MI FINAL. AS WE CROSSED THE RWY; WE OBSERVED THE TURBOPROP ON THE GAR. THERE WAS PLENTY OF ROOM FOR THE AIRPLANE ON FINAL TO LAND. UPON CHANGING TO TWR; THEY TOLD US WE WERE NOT CLRED TO CROSS. THE FO STATED THAT WE WERE CLRED TO CROSS BY GND AND HAD VERIFIED OUR CLRNC PRIOR TO XING AND THAT ALL 3 OF THE COCKPIT CREW UNDERSTOOD THAT WE WERE CLRED TO CROSS. I HAVE NO SUGGESTION ON HOW TO PREVENT A RECURRENCE EXCEPT TO SAY THAT LANGUAGE AND ACCENT MAY HAVE BEEN A BARRIER. SUPPLEMENTAL INFO FROM ACN 519174: ON TAXI OUT; WITH FO ON RADIO; GND CTL'S ORIGINAL INSTRUCTION WAS FULL LENGTH OF RWY 25R VIA OUTER 8 AND E5 TO HOLD SHORT OF RWY 20. AFTER 1 ACFT EXITED RWY 20 AT E4; GND REVISED INSTRUCTIONS TO E4. I THEN UNDERSTOOD CLRED TO CROSS RWY 20 AT E4 AND CALL TWR ON 118.6. FO ASKED AGAIN; AND AGAIN I UNDERSTOOD WE WERE CLRED TO CROSS RWY 20. PRIOR TO XING; THE FO STATED 'CLR R.' I AND CAPT LOOKED L. TFC WAS ON FINAL FOR RWY 20; BUT AMPLE TIME TO SAFELY CROSS RWY 20 EXISTED. AS WE CLRED RWY 20 ONTO F4; ALL 3 CREW MEMBERS NOTED THE TFC ON FINAL TO RWY 20 COMMENCING A GAR. AFTER A FEW MINS ON 118.6; THE TWR CALLED US SAYING WE HAD NOT BEEN CLRED TO CROSS 20. FO REPLIED WE HAD. TWR AGAIN SAID WE HAD NOT AND HAD CAUSED A GAR. I WOULD LIKE TO HEAR THE TAPES. I UNDERSTOOD; TWICE; WE HAD A XING CLRNC.