Heater blows cold air in winter, AC blows hot air in summer. AC compressor never engages. What and where should I examine? Help.

a blown fuse would be the very first thing to look for. do the climate control levers operate smoothly and stop where you believe they should? if not, there may be something broken behind the control panel.
it could be that a fresh thermostat will solve your heating problems and you should probably confirm that your a/c system has a full charge.