I'm having trouble getting continuous motion from my winding machine.  When I run this snippet of code, the machine moves in discrete steps, coming to a complete stop after every line.  This is making things terribly slow.

G91 (Relative positioning)
G1 A120 C1080  F10
G1 A120 C-1080 F10
G1 A120 C1080  F10

Curiously, when I add a linear axis, the machine behaves as I would expect.  The A axis rotates continuously, without stopping.

G91 (Relative positioning)
G1 X1 A120 C1080  F10
G1 X1 A120 C-1080 F10
G1 X1 A120 C1080  F10

I've already tried adding a dummy "X0" command, but it doesn't help.  What can I do?

Have you got Exact Stop turned on?  That would cause an axis to decelerate to 0 between each move.

Sounds like it. Change this in Config>General config or just insert a G64 command in the start of your program.

This makes sense, but for whatever reason it doesn't work.  I turned on the constant velocity mode, and added G64, but it behaves exactly the same way.  What else could it be?

Check your initialisation string just over the constant velocity/exact stop checkbox. If there's a G61 in there swap out for a G64. If this doesn't fix it I'm stumped

There's G80 and S500.  Still no luck.

Could this have anything to do with the Apollo III controller I'm using?

Could be but if so I'm in unknown territory. With that said it should honour modal commands from Mach along with all the other commands.