Small Arms Training, Volume I, Pamphlet No.13, Grenade, 19xx, makes something of a meal of left and right handed throwing.
The 1942 version, when dealing with instruction for assembly of the No.36 states, "If the thrower is right-handed, replace pin from right to left, if left handed from left to right."
The 1951 Pamphlet instructs similarly and has photos of both right handed and left handed throwers; and again the 1964 Pamphlet likewise, but with one photo (labelled as right handed).
Clearly, changing pin orientation was done prior to insertion of the separate igniter set. With the advent of the L2 (1975 Pamphlet), changing over the pin was expressly forbidden for obvious reasons.
Of course, that's the training manual. In practice what happened was in some cases another matter. Troops often did not prime their own grenades but had them issued by the platoon sergeant, with orders not to mess around with them. My father - who was left handed - was issued with primed 36 grenades many times in the Sicilian and Italian campaigns in WWII, and just threw them as was. Pulling the pin with the "wrong" hand proved not to be so difficult after all.
Tom.