[xnyan]

Don’t do this, a cheap non-programmable thermostat is a better option and inexpensive.

[Scoundreller]

Wire them both up so that your cheap thermostat provides hard upper and lower limits.

Your HVAC doesn't care where it gets a 24V signal. An open relay won't mind being energized from the "wrong" direction. AFAIK, thermostats won't tattle.