GIV CAPT REPORTS RWY INCURSION AND CRITICAL CONFLICT DEPARTING COE RWY 01.

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WE WERE DEPARTING COE ON AN IFR FLT PLAN. THE WINDS WERE OUT OF THE N (020 DEGS) AT 4 KTS ACCORDING TO ASOS THE ACTIVE RWY WAS RWY 1. WE STARTED ENGS AND ANNOUNCED OUR INTENTION TO TAXI TO RWY 1 FOR DEP. HOLDING SHORT OF RWY 1; WE CONTACTED SPOKANE DEP TO GET OUR CLRNC. AFTER A SHORT DISCUSSION WITH DEP CTL ABOUT KNOWN INBOUND TFC WE RECEIVED A CLRNC TO DEPART AND MAINTAIN VFR TO 12000 FT. WE WERE ADVISED BY SPOKANE DEP CTL TO WATCH FOR A HELI INBOUND FROM THE SW. WE ANNOUNCED OUR INTENTION TO TAKE OFF ON CTAF OF 122.8 AND I CALLED FOR THE LINE-UP CHKLIST. I CHKED FOR TFC ON BASE AND FINAL FOR RWY 1 AND ALSO FOR THE HELI INBOUND FROM THE SW. AS WE PROCEEDED TO TURNABOUT 30 DEGS R TO TAXI INTO POS WE SAW A LIGHT TWIN JUST LIFTING OFF FROM RWY 19. IT PASSED OVER US AT ABOUT 100 FT. ALTHOUGH WE LOOKED TO OUR R TO ENTER THE RWY WE NEVER SAW THE OTHER ACFT UNTIL IT LIFTED OFF. THE LAST THING WE EXPECTED WAS TO SEE ANOTHER ACFT COMING DOWN THE RWY THE WRONG WAY. THAT MAY BE PART OF THE REASON WE DIDN'T SEE THE OTHER ACFT UNTIL IT LIFTED OFF. IT IS DIFFICULT ENOUGH TO AVOID RWY INCURSIONS AT CTLED ARPTS LET ALONE UNCTLED ARPTS. IT WAS OBVIOUS TO EVERYONE THAT RWY 1 WAS THE ACTIVE RWY WITH TOUCH-AND-GOES GOING ON AND OTHER ACFT TAKING OFF JUST BEFORE US. IT IS VITALLY IMPORTANT AT UNCTLED ARPTS THAT EVERYONE COOPERATES WITH THE TFC FLOW IN PROGRESS. OUR EXPERIENCE AT COE CERTAINLY JOGGED OUR MEMORY ABOUT EXPECTING ANYTHING AND EVERYTHING AT AN UNCTLED ARPT.