If you have the spare contact pair, sure. But it's a little odd that you
have a spare pair. Perhaps a 120V/240V convertible saw wired for 120V.

No, I think he just used something surplus. It is a 4 pole 120v relay, max
1hp. Since the motor is 2hp he jumpered two of the poles and used both.
One more pole is used to maintain current to the relay after you let go of
the the start switch. That leaves one extra pole. Since only one of the
original two poles are fused (or have a broken spring), I was able to use
the extra as a replacement.

I would have to pull the whole thing out to even look at it; just changing
the wiring was not easy. I will save pulling it out for when I have to
replace it.

Since you know about relays, let me ask you another question. This is a
$100 potts and something or other. In the McMasterCarr catalog they have a
240v (if I have to replace it I will rewire from 120v to 240v) 25a relay for
$12. Will something that cheap actually work? The motor is 8a at 240v, so
it is well within the 25a limit, except at start, where it is probably 28a
or so. Will that be okay?

Thanks.