What would cause a home window air conditioner to blow cold air out the back and warm out the front?

Question:

Answers:
Geez,,it's been Forever since I fooled with a window unit.

They usually have 2 Modes of cooling,,
1)Fresh Air ,,,,which takes in OUTSIDE air,,,cools it,,and blows into Room.
and
2)Recirculate,,,,which takes INSIDE air then "loops" it thru the cooler

That's done with a Blend Flap/or Door.

When you get a condition like what Your describing,,,
It's usually a matter of that Door being "Stuck" in a wrong position.

It ends up blending Warmer outside Air with Heated Air that Should be Exhausted outdoors,,,,and send all that INDOORS.

The Cool air being exhausted to Outside is being sorta "siphoned" by the Exhaust Airflow.
It's the Cooled air supposed to be Diverted to INDOORS,,,but is Blocked by the mispositioned Blend Door.

It's a fairly Simple & straightforward mechanism.

You "SHOULD BE" able to pop Front Grille Panel Off the AC,,,and see that flap,,,and unstick it,,,or push it into position.

Sorry I cant explain specifically,,,
but it's all pretty obvious when you see it.

Find the lil' Maze/Duct Work inside the AC Case,,,
then look for a Crank Arm with a Cable or Rod used to Move it.

That crank arm is attached to the blend Flap/Door inside the Duct.

Manually move it and monitor the airflow.

Sometime pieces can break,,,
Or a seal or gasket (just some weatherstripping type foam usually)on the Door or Cases can jam it's movement.

Sounds Odd,,but I've seen Mice nests in them,,
and Wasp or Mud dawber wasps,etc nests in them.
And blocks the door from closing to correct position.

You should be able to dismantle the duct fairly quick and easy to check for Full Opening,,Closing,,,and any causes of problems in the duct work/blend Flap

Good Luck.

* Be Careful Poking around in there!
They have Capacitors which Store Electricity even when AC is UNPLUGGED,,,which can give a serious Shock.

That is normally NOT a problem.
But DO NOT take for granted all the wires are "SAFE" just 'cuz unit is Unplugged