Does connecting a battery charger the wrong way damage the car electrical system?

If there is any charge left in the battery, connecting it the wrong way should blow the fuse in the charger before any damage is done.
Motors, heaters and lamps will not be affected.
Electronics, such as radiOS, cds, and engine management systems however, do not like reverse polarity and will blow the semiconductors inside, if reverse connected.
Most properly designed and constructed car electronics, have protection on their power supply lines, to stop devastating destruction caused by reverse polarity. This is usually a reverse connected diode across the power input lines.
Connected properly, the diode does not conduct and has no effect. Conncted in reverse and the diode will conduct, putting a short across the line and hopefully blowing a fuse to protect it. (crowbar protection).
So, you may get away with it. But it is not advised.