According to Toller :
My table saw has an relay on it, controled by an on/off switch.
Today it wouldn't turn off unless I unplugged it. When I plugged it back
in, it ran immediately.

Welded contacts most likely. Relays like this are often a single
"activator" with multiple spring-loaded contacts. One pair of contacts
has welded closed.

Or could be broken spring, a piece of grit or something.

I guess the real question is, is using a different pole a reasonable thing
to do?

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.

That being said, in some relays a stuck contact pair _may_ begin to
interfere with relay operation. Stretching springs a bit much, distorting
activator movement, increased likelyhood of jamming etc.

If possible, I'd try to free up the bad contacts to prolong the lifetime of
the relay (even if you're not going to switch back to the burnt contacts).

If you can't open the relay housing, give it a good sharp rap with screwdriver
handle. If you have a VOM, attempt to test whether you've managed to free
the contacts. Don't worry if you can't, it may be a broken spring (which means
you probably can't get them to stay apart).

If you can open the relay housing, examine the contacts, and pry them apart
if stuck. Don't bother putting the wire back on the contact unless you
can polish the contacts - ie: with an emery board (and there's enough metal left
on the contacts).

Generally speaking there's no hazard in continuing to use the relay even
if the unused contacts remain fused other than it may refuse to stop again
in future.
--
Chris Lewis, Una confibula non set est
It's not just anyone who gets a Starship Cruiser class named after them.