As far as I can gather there are some H sets that can only be run in push-pull mode.
The August 27th timetable change is rumored to see the P class loco retired therefore no push-pull sets.
From what little knowledge I have about their wiring I'd suggest the H sets are going to SSR to have some work done so the N class can haul them.
All H sets can be run normally and don't need to be in push-pull.
The HEP units on the P's aren't capable of supplying power to 6 H cars so when you see them in push-pull mode each loco is only supplying power to the 3 car set it is attached to, with the HEP cable disconnected between the two sets. I haven't checked to see if the six car SSH sets have the HEP cable permanently connected between all vehicles but I suspect that would be the case.
N class has an 'issue' the H sets when the MU cable is connected to them as some of the H's circuitry is shared with the dynamic brake which means the dyno can't be used, but other wise there is no problem hauling them with an N.