CL65 CREW AND A TWIN CESSNA HAD A TFC CONFLICT IN ZLC CLASS E AIRSPACE.

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OUR FLT DEPARTED BZN AT XA32 OFF OF RWY 12. WE WERE INITIALLY GIVEN AN ALT OF FL310 IN OUR CLRNC AT THE GATE. AT THE END OF RWY 12 WHEN THE TWR CLRED US FOR TKOF; WE WERE GIVEN AN AMENDED ALT OF 17000 FT. THE TWR DID NOT CLR US ON COURSE. WE UNDERSTOOD THAT WE WERE TO FLY RWY HDG. THE TWR TOLD US TO CONTACT ZLC. I DID THIS; RPTING CLBING THROUGH 5700 FT FOR 17000 FT. THE CTLR THEN GAVE US FL310. AT NO TIME DO WE REMEMBER BEING GIVEN AN ON-COURSE DIRECTIVE. APPROX 15 MI FROM THE ARPT (BZN) ON THE HDG OF 120 DEGS; THE CAPT NOTICED TFC ON THE TCASII AT OUR 12 O'CLOCK POS DSNDING -- IT WAS APPROX 7 MI AWAY AND AT 15600 FT. HE INSTRUCTED ME TO ASK CTR FOR 'ON COURSE.' HE SAID WE SHOULD HAVE BEEN ON COURSE 20 MI AGO. HE THEN INSTRUCTED US TO MAINTAIN 13000 FT (WE WERE ALREADY CLBING THROUGH 13800 FT; SO WE HURRIEDLY WENT TO 13000 FT). HE THEN GAVE US A COURSE CLRNC NOT ORIGINALLY ON OUR FLT PLAN. WE COMPLIED. THE CTLR THEN INSTRUCTED US WE WERE INVOLVED IN A TFC CONFLICT WITH A TWIN CESSNA AND THAT HE WOULD PULL THE TAPES. I BELIEVE HE THOUGHT WE SHOULD HAVE BEEN CLRED ON COURSE ALREADY AND DID NOT NOTICE THE TFC SIT. I THINK THERE WAS A COM BREAKDOWN BTWN THE TWR AND CTR WHICH RESULTED IN ONE WITH US. WE NEVER HAD A TA OR AN RA ON THE TCASII DURING THE EVENT. I BELIEVE THAT THE COM BREAKDOWN BTWN ALL PARTIES AND FATIGUE OF THE CREW CONTRIBUTED TO THE CONFLICT.