B727 CREW HAD ATCT GND CTLR ISSUE CLRNC TO RWY 35 AT SLC. CTLR THEN EXPRESSED SURPRISE THE CREW CROSSED RWY 34R ON THE WAY TO RWY 35.

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THE FO CALLED SLC GND CTL FOR A TAXI CLRNC FROM A CARGO AREA NEAR THE APCH END OF RWY 34R (H2 TXWY. THE CTLR ASKED IF WE COULD ACCEPT RWY 35; THE FO SAID YES; AND THE CTLR CLRED US TO; 'TAXI TO RWY 35.' I TAXIED S TO THE H1 TXWY; TURNED L; AND CROSSED THE RWY 34R APCH END. I HAD TURNED ON ALL TAXI LIGHTS BEFORE I CROSSED THE RWY AND BOTH THE FO AND I LOOKED FOR TFC; BUT WE DID NOT SEE ANY ACFT. AFTER WE CROSSED RWY 34R THE CTLR ASKED; 'DID I CLR YOU TO CROSS RWY 34R?' WE BOTH LOOKED AT EACH OTHER AND AGREED THAT WE HAD HEARD THE CLRNC AS 'TAXI TO RWY 35.' THE FO CALLED BACK; 'WE BOTH HEARD; CLRED TO TAXI TO RWY 35.' NOTHING MORE WAS SAID AFTER THAT EXCHANGE OF RADIO CALLS. IT WAS THE LATE NIGHT SHIFT WITH VERY LITTLE TFC MOVEMENT. THE CTLR WAS WORKING GND; TWR; AND APCH. ONE ACFT WAS INBOUND TO LAND ON RWY 16L (SAME RWY AS RWY 34R). WE TOOK OFF ON RWY 35 AFTER THE OTHER INBOUND ACFT LANDED. THERE WERE NO OTHER ACFT IN THE AREA. I SUSPECT THAT THE CTLR WAS OVERLOADED OR DISTRACTED. HE MAY HAVE THOUGHT THAT WE WERE E OF RWY 35 INSTEAD OF W OF RWY 16L. THERE DID NOT APPEAR TO BE ANY TFC CONFLICT.