CL601 FLT CREW CLBS FROM ASSIGNED ALT WITHOUT CLRNC ON AN OCEANIC RANDOM RTE.

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MY COPLT AND I WERE FLYING AN EMPTY ACFT FROM GANDER (CYQX) TO ZZZ. OUR INITIAL CLRED ALT WAS FL290. WE HAD FILED TO CLB TO FL370 AFTER N5900 W04000; WHICH WAS AFTER ABOUT 2 HRS INTO OUR FLT. AFTER MAKING OUR POS RPT AT N59 W040; I MADE SEVERAL ATTEMPTS TO CONTACT GANDER RADIO TO CLB. I CANNOT RECALL MY WORDING FOR MY REQUEST TO CLB FROM FL290 TO FL370; THE FREQ WAS VERY CONGESTED. I BELIEVE I SAID CHALLENGER WOULD LIKE TO CLB FROM FL290 TO FL370. GANDER RADIO RESPONDED WITH CHALLENGER RPT REACHING FL370. THEN I SAID CHALLENGER IS LEAVING (OR IS OUT OF) FL290 FOR FL370. JUST AS WE LEVELED OFF AT FL370; GANDER RADIO REQUESTED OUR ALT. I RESPONDED THAT WE WERE LEVEL AT FL370. THEN GANDER RADIO SAID ATC IS REQUESTING (OR EXPECTING) YOU AT FL290. I ASKED IF WE SHOULD DSND TO FL290. THERE SEEMED TO BE SOME CONFUSION AND HE SAID TO STANDBY. HE RETURNED AND CLRED US TO FL290. AFTER REACHING AND RPTING FL290; WE WERE ASKED IF WE HAD BEEN CLRED TO CLB TO FL370. HAVING NOT REMEMBERED HEARING THE WORDS 'CLRED TO;' TRIED TO EXPLAIN HOW WE HAD BEEN LED TO BELIEVE THAT WE HAD BEEN CLRED TO CLB. WE ASSUMED THAT SINCE WE HAD BEEN TOLD TO RPT REACHING FL370; THAT WE HAD A CLB CLRNC. OUR FILED CLB TO FL370 WAS DUE TO FUEL PLANNING FOR THIS LONG FLT. WE WERE NEEDING THE CLB TO BE ABLE TO COMPLETE THE FLT. WE WERE GETTING ANXIOUS FOR CLB CLRNC AND READILY ACCEPTED WHAT WE THOUGHT WAS A PROPER CLRNC.