PREFLT PROC DEICING HAS TO BE REPEATED WHEN THE FIRST PROC WAS FOUND TO BE INCOMPLETE. ALERT DEADHEADING CABIN ATTENDANT AND A PAX NOTED SNOW; SLUSH AND ICE ON TOP OF BOTH WING FILLETS.

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DFW RWY 36R THRESHOLD DEICING PAD. WE HAD WAITED IN LINE FOR THRESHOLD DEICING FOR 3 AND ONE HALF HRS. AFTER POSITIONING OUR ACFT IN THE PAD; A CERTIFIED DEICER ASKED ME TO PREPARE THE ACFT FOR DEICING. I TOLD HIM THAT I WANTED THE ENTIRE ACFT DEICED AND TO CHK THE ENG INLETS FOR CONTAMINATION. AFTER ABOUT 20 MINS HE AGAIN RETURNED TO THE INTERPHONE TO ADVISE ME THAT THE ACFT HAD BEEN DEICED AND WAS CLEAN; WITH A SUBSEQUENT CLRNC TO START THE ENGS. AFTER STARTING THE L ENG; I WAS CALLED BY ONE OF THE FLT ATTENDANTS AND WAS TOLD THAT A PAX AS WELL AS A DEADHEADING FLT ATTENDANT HAD NOTICED THAT THE WINGS WERE STILL COVERED WITH SNOW AND ICE. WE SHUT DOWN THE ENG AND RECALLED THE DEICER TO THE ACFT. I TOLD HIM THAT THE ACFT WAS NOT CLEAN AND TO RESPRAY THE WINGS. I ALSO HAD THE FO PERFORM A CABIN INSPECTION AND HE ALSO CONFIRMED THAT THE INBOARD THIRD AND AFT HALF OF THE WING CONTAINED LARGE AREAS OF SNOW AND ICE. AFTER THE CERTIFIED DEICER AGAIN STATED THAT THE ACFT WAS CLEAN; A CABIN CONFIRMATION WAS DONE. HAD IT NOT BEEN FOR AN OBSERVANT FLT ATTENDANT AND AN EDITOR FOR A MAJOR MAGAZINE; OUR TKOF MIGHT HAVE BEEN JEOPARDIZED. I BELIEVE A CABIN INSPECTION SHOULD BE A MUST ANY TIME DEICING IS ACCOMPLISHED. IT IS VERY APPARENT THAT STANDARD PHRASEOLOGY OF 'YOUR ACFT IS CLEAN' BY A CERTIFIED DEICER DOES NOT INSURE THE LEVEL OF SAFETY REQUIRED.