I'll put my engineer's hat on and answer your original question, electrically anyway. My PC is rated at 3.7 amps. Taking current X 120VAC = 444 watts. If the unit was rated at 500 watts continuous it would in theory provide enough power. Most inverters' packaging advertises their peak power in large print (500W) and then in fine print they mention continuous draw (250W). If that is indeed the case, then the unit will not have enough power.
The other problem is the connection to your battery. I have seen inverters in this range with both a cigarettte lighter and alligator clamps. Most cigarette lighters are fused at 20 amps or 20A x 12VDC = 240 watts. 444 watts off a nominal 12 VDC system would draw 37 amps - you'd pop the fuse the first time the motor was loaded down.
The inverters are not 100% efficient (they generate a fair bit of heat as wasted energy), further complicating the selection of an inverter.
If you are set on using an inverter for the occasional job, I would get one that is closer to 750 watts continuous (1250 to 1500 watts peak) and hard wire it to your car battery. Most at this level have built in fuses for circuit protection, reverse polarity protection, and low voltage shutdown (the inverter would shut down before it ran your car battery to low to crank). Of course, this eliminates the $20 inverter that sparked your interest.
The 3.7A / 120 VAC rating on the PC nameplate is probably with the unit loaded down to near stall condition where it would draw the most current. If you are using it for light polishing or applying glazes, sealants, or waxes then you would not likely draw that much current. I think the bottom line is that if the unit is rated for 500W continuous, it was hard-wired to the battery and you weren't heavily loading the PC motor it would work. For maximum flexibility I would step up to a 750 W unit.
My brother is a trim carpenter and has a 1500W continous inverter hard wired to his pickup for the rare situations where he cannot get to 120 VAC on a job site and has yet to have any electrical problems.