AT PLT REQUEST; CTLR ISSUES CLB THROUGH OCCUPIED ALT RESULTING IN LTSS.

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ENRTE AT FL330 EBOUND ON J42 100 NW W OF DFW. WE REQUESTED FL370. CTLR INITIALLY SAID UNABLE BUT WOULD KEEP US ADVISED. HE CALLED BACK AND CLRED US TO FL370; BUT AS WE PASSED FL340 HE TOLD US TO STOP AT FL350. WHEN WE ARRIVED AT FL350; HE TOLD US TO CLB IMMEDIATELY TO FL370 AND BE LEVEL IN 1 MIN. I ACKNOWLEDGED AND FO STARTED CLB. WE OBSERVED A HEAD-ON ACFT AND CTLR CAME BACK AND ISSUED EVASIVE TURNS TO US BOTH (TO THE R) AND TOLD US TO STOP CLB (WE WERE AT FL360). BOTH ACFT HAD 4 DIGIT CALL SIGNS WITH ONLY THE SECOND DIGIT DIFFERENT. I WAS VERY CONCERNED THAT EITHER WE OR THE OTHER ACFT HAD ACCEPTED THE WRONG CLRNC. I WAS ALSO CONCERNED THAT I MIGHT HAVE MADE A MISTAKE LEAVING THE EVASIVE FLYING TO THE FO. SUBSEQUENT REVIEW OF THE COCKPIT VOICE RECORDER PROVED HOWEVER THAT BOTH ACFT RESPONDED TO THEIR APPROPRIATE CLRNCS AND IRONICALLY; HAD WE BEEN ABLE TO CLB FASTER; WE MIGHT HAVE HIT. THE OTHER ACFT PASSED AT OUR NEWLY ASSIGNED ALT OF FL370; SO OUR WT AND TURN LIMITED RATE OF CLB SAVED OUR LIVES.