A tub filler spout with a diverter is installed in my spa tub. It doesn't have a shower. I want to add a hand-held shower, but the diverter prevents me from doing this (it is one of those pull-down from the spout types). How do I remove the spout, twist or pull? I have gently tried both, but it is very sluggish. I don't want to damage the pipe connection.

As you said... tub filler either twists off counterclockwise or there will be an allen screw located at bottom/back (and indent) of tub spout.

The few of these that I have done actually spun off... and then there was a unit that actually had the allen screw under the spout... I had to spin off tub spout to find the allen screw so I could replace with a new spout.

Look for an allen screw under spout (use compact mirror if needed). If no allen screw then just grab spout and twist counterclockwise until loosened... Like I said, then you should see an allen screw into a threaded and o-ring type adapter that will need to be removed to install a new diverter spout.

Check out the tub diverter spouts in pic. Below... should help with the new hand-held shower you want to install!

Let us know what you find... Mark

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Mark gave you excellent instructions, perhaps a picture or two might help.
The two common types o0f spouts are, slip on and threaded, (see images) As Mark said, if it's a slip on simply roll it over and loosen the Allen Head set screw and pull it off. This spout will be chrome over plastic while the4 threaded will be chrome over brass. But the diverter I think you'll be interested is the diverter with a hand held shower connection such as the one Mark put up. Here's another peeki at one,(see image) Good luck, Tom