This whole "twist the wires" together thing is somewhat bogus to begin with. It only works due to an academic curisity (N-channels tend to be stronger than P-channels on the same process, that or some sync outputs are open-drain to being with) anyway. The proper way to make composite sync is to XOR the two sync signals together, minding polarity. If you take active low ("negative") horizontal sync and XOR it with active high ("positive") vertical sync, you get negative composite sync. Hooking two CMOS outputs together is actually a kinda bad thing to do and tends to eventually result in dead output drivers (though in practice that seems to take a rather long time).
Why did I bring all that up? Well, if you do it "the right way" using a 7486 (or similar: 74LS86, 74HCT86, 74F86, 74S86), you'll have both composite sync (at the output) and separate sync (at the inputs) available. This way, you can feed your monitor and light gun what they want.