Are there reference speeds like V1, Vr or V2 for a touch and go procedure?
While for touch and go there isn’t a speed that carries the same guarantees as the takeoff speeds (you will not hit anything nor will you exit the runway on ground at undue speeds), for procedure reasons there are indeed predefined speeds at which to rotate and at which to climb. On the 777, for example, rotation is called at VREF (standard manual approach speed for selected flap setting) on touch and go, and climbout is flown at VREF+15 to VREF+25. . 

In any case, the touch and go manoeuvre is only a training manoeuvre and is not in any way considered in performance calculations. Since it’s not calculated, it is also not a manoeuvre for which sufficient runway distance and/or obstacle clearance is assured by either calculation or regulation, neither on all engines nor following an engine failure, other than by flight crews’ experience and prudence. The actual speeds for touch and go are therefore somewhat more arbitrary than on takeoff.

The closest manoeuvres that are considered in performance calculations are a normal takeoff and a go-around from minimums. The relevant speeds for the former you know, for the latter it is the predetermined approach speed.