Any suggestions for incorrectly-wired thermostat?
Since you have baseboards, you don't have a fan.  That means it's probably a 2-wire thermostat.  Check the voltages with a voltmeter to make sure  you're not playing with 120/230V line voltage.  (it's likely either 24V or millivolt). 

The thermostat you describe is for a 24-volt system. Those simply short wires together.  To call for heat, they short R, G and W.  To call for air conditioning, Y, G and W.  If it shorted all 4, it would run heat and A/C against each other, which would dry out the air in a hurry.  (your car's "defrost" mode does exactly this).  Here's the upshot: If you short random wires together, nothing terrible will happen. That should remove some fear. 

If your thermostat is millivolt, it either shorts the wires, or does not.  Again, shorting will do no harm and probably cause the heater to fire. 

Your system doesn't have a fan, so G is irrelevant.  You don't have A/C, so Y is irrelevant.  So, if you short the right two wires together, the heat should fire.  That's it. 



Here's the only wiggy bit:  They doubled up the wires.  That's because they're using telephone cable which is maybe too thin for thermostat service.  You should probably double them too.  How do you find the right pair?  

First, find the right singles.  Short wires 1 and 2 and see if the furnace fires.    Don't do it for any longer than it takes to find out.  Next, try wires 1 and 3.  Does it fire?  OK, then, logic it out.  It must mean 2 and 3 are equivalent.  Try every combination. If you get it wrong, the heat will run continuously. 

Once you have them paired, attach one pair to "24VAC" (R) and the other to "Heat call" (W) on the thermostat.  Doesn't matter which one, since the thermostat just shorts them either way. 

(it will matter on a thermostat with 3 or more functioning wires, be it Fan (G) or common (C) for a smart thermostat.)