Curt,
Opinions on this topic vary widely and as Frank notes, there have been lots of heated discussions.
I drove my 58 (registered as a 59) TR3A to work most days for over 20 years (estimated 200,000 miles), and this combination worked very well for me :
Engine : Valvoline 20W50 SYNpower (full synthetic)
Trans : Valvoline 20W50 Racing oil (conventional, recently relabeled "Not Street Legal")
Diff : Valvoline 75W90 SYNpower gear oil (full synthetic)
I believe Redline MT90 (which is a full synthetic) is also suitable for the gearbox, but don't have enough miles on it to recommend it. Without OD, IMO the choices are wider but I would stick with a full synthetic rated for manual transmission use (MT-1) (like the SYNpower or Redline MTL).
The GL4 vs GL5 discussion focuses on an "extreme pressure" additive that is typically found in both GL4 and GL5; but most conventional GL5 oils have a lot more of it than conventional GL4 oils. This additive can cause the brass thrust washers used in Triumph differentials to erode over time. This is the same additive that gives used gear oil that distintive smell : as the additive breaks down it forms sulfur compounds that are both very smelly and harmful to copper alloys. However, as I understand it, synthetic oils have a higher film strength to begin with and so require much less of the additive. At any rate, the diff thrust washers in TS39781LO were already pretty bad when I got the car in 1982 and never seemed to get any worse in all those hard miles using the GL5 synthetic.
Also worth noting that there is nothing in the GL4 standard that limits how much of the additive can be present; in fact it's possible to formulate a conventional oil that meets both GL4 and GL5 specifications (but AFAIK it's no longer marked that way).