FLC OF A DC10 MISUNDERSTOOD ATC AND DSNDED TO LAST ASSIGNED ALT DURING A DEP CLB RESULTING IN A TCASII ALERT OF OPPOSITE DIRECTION TFC AT THEIR SAME ALT. ATC INTERVENED AND INSTRUCTED THEM TO IMMEDIATELY CLB AS INSTRUCTED IN A PREVIOUS CLRNC PRIOR TO A TA WHICH THEY HAD MISUNDERSTOOD TO DSND.

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ON CLBOUT FROM RIO WE WERE ASSIGNED PORTO 3 DEP. DURING CLBOUT WE WERE GIVEN 'FLY HDG 000 DEGS AND MAINTAIN FL110' BY ATC. WE WERE THEN CLRED DIRECT TO ASGA WAYPOINT. A SHORT TIME LATER WE WERE AGAIN INSTRUCTED TO 'FLY HDG 000 DEGS AND LEAVING FL250 CLRED DIRECT TO ASGA.' AS WE PASSED FL250 ON THE CLB TO FL280 WE INITIATED A TURN (L) TO ASGA. WE ADVISED ATC OF OUR ACTION AT WHICH TIME WE WERE TOLD TO MAINTAIN HDG (000 DEGS IN BRAZILIAN TERMINOLOGY; 360 DEGS IN USA TERMINOLOGY) AND CLB TO FL280. ATC THEN ADVISED US '...LEVEL AT FL260' AS WE WERE PASSING FL265. WE INITIATED AN IMMEDIATE DSCNT TO FL260 AND ADVISED ATC OF OUR ACTION. AT THIS POINT A TARGET APPEARED ON TCASII AT 12 O'CLOCK POS AT AROUND 15 NM AT OUR LEVEL; OPPOSITE DIRECTION. ATC QUERIED WHY WE WERE AT FL260 AND INSTRUCTED US TO CLB. WE INITIATED AN IMMEDIATE 2000 FPM CLB AND 30 DEG TURN TO R TO AVOID A TFC CONFLICT AND ADVISED ATC OF OUR ACTIONS. THE TCASII TARGET PASSED 10 NM L OF US AT FL260 (2000 FT BELOW). WE WERE SUBSEQUENTLY CLRED ON COURSE. ATC QUERIED WHY WE HAD DSNDED TO FL260. OUR RESPONSE WAS THAT HE HAD CLRED US TO FL260 WHICH HE VEHEMENTLY DENIED. BOTH PLTS HAD HEARD THIS INSTRUCTION. ON REFLECTION LATER WE CONCLUDED THAT THE INSTRUCTION TO MAINTAIN FL260 COULD POSSIBLY HAVE BEEN AN AIR TA WITH THE FIRST PART OF THE XMISSION GARBLED. BRAZILIAN ATC HAVE THE HABIT OF ADDING YOUR CALL SIGN BEFORE AND AFTER A MESSAGE. IN ANY EVENT WE CAME NO CLOSER TO THE TFC THAN 10 NM AND 2000 FT VERT SEPARATION.