ARTCC RADAR CTLR TRANSPOSED ACFT CALL SIGNS AND CLRED WRONG ACFT TO CLB. LTSS OCCURRED DUE TO MIXUP.

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SHORTLY AFTER BEING CLRED TO OUR DIVERT ARPT (CHEYENNE) A TFC CONFLICT OCCURRED. WE WERE CLRED TO TURN R TO 055 DEG HDG AND CLB TO FL270. OUR CALL SIGN WAS ACR X ABC (IMMEDIATELY BEHIND US WAS ACR Y BAC). AFTER ACKNOWLEDGING AND COMMENCING TURN AND CLB TO FL270; I WENT OVER TO TALK TO MY COMPANY DISPATCH WHILE MY COPLT COMMUNICATED WITH ATC AND WAS FLYING THE AIRPLANE. DURING THE CLB/TURN THE CTLR REALIZED HE HAD TRANSPOSED THE CALL SIGN NUMBERS IN HIS MIND AND DID NOT MEAN FOR US TO CLB. APCHING FL270 HE CALLED FOR US TO TURN AND DSND IMMEDIATELY TO FL260. AS WE APCHED FL260 WE GOT AN RA ON OUR TCASII TO CLB. MIX UP AND CONFUSION AS TO 'WHO' WAS BEING GIVEN 'WHAT' CLRNC; RESULTED IN A TFC/ALT CONFLICT. PROCEEDED WITH NO FURTHER INCIDENT. PHONE CONTACT ON THE GND WITH ATC SUPVR AT ZDV REVEALED THAT OUR CLRNC TO CLB WAS GIVEN MISTAKENLY BY THE CTLR. THIS IS ALL ANOTHER EXAMPLE OF HOW IMPORTANT IT IS TO MAINTAIN CONSTANT VIGILANCE; ESPECIALLY WHEN 'CLOSE SOUNDING OR TRANSPOSABLE CALL SIGNS ARE ON THE SAME FREQ; WHEN COMMUNICATING.