Hi Brian: First of all never say it isn't the lid switch, jump it out and try the machine. I have seen bad switches that worked intermittenly.
2/ check the hoses cold to cold and hot to hot, if backwards it will not rinse and there fore not advance to spin cycle.
3/ check to see if only cold water is entering the washer, I have had people tell me that the washer fills and washes but will not spin, but they had their water temp set to warm or hot so it did fill but rinse is only cold and there fore no cold water would enter the washer for rinse hence no rinse no spin.
4/ check the drain hose. is it sealed in the drain line, if so it can syfin all the water out of the washer during the rinse cycle. The drain needs an air vent in it to prevent this. some people tape the hose in the drain line and it may work the best kind for a while but for some reason the bit of air it was getting gets blocked and then it syfins the water out. the washer will never fill for rinse so it dose not reach the spin cycle.
5/ check the coupling between the motor and trans I have see them cracked and they would turn the agaitor but not spin as the load is greater in spin
6/ lastly it could be the clutch or brake release start with the easy stuff and work your way through the rest if need be
Hope this helps you out