You’ll find a lot of information in the link above, as well as at many of the plastic suppliers and sources like McMaster-Carr, Curbell, or Tap. All have selection guides that cover materials they sell, and in some cases, a broader range.
(read this with a grain of salt and do your own research, as so much depends on application and design. I have some experience here, but am not an expert. If you can tell us more about the application, you will get better information. Things such as rough design, loading, allowable deflection, etc)
Without further information about what you are doing, your “VERY rigid” requirement limits you a lot. HDPE isn’t, PVC may be fairly stiff, depending on the blend, ABS and Polystyrene again depend on the blend, but are only generally not very stiff. Polycarbonate is ok in this respect, and tough, glass filled polycarb being quite good. Nylons can be pretty much anywhere from moderate E to pretty stiff, for a thermoplastic. Acetyl (Delrin) is pretty good as an engineering material overall, and is generally about the same as nylon stiffnedd-wise.
The glass reinforced polymers are much stiffer, but have their own issues in machining. Glass reinforced nylon and Acetyl are very good, and glass reinforced polycarbonate is awesome, and even glass reinforced PTFE (teflon) isn’t bad. The problem is that you need really good dust and chip control, as the machining process will produce glass fibre dust that is very, very harmful, health-wise and to the machine, if not controlled.
You also have the thermosets. A filled castable epoxy may be a good choice if you need very stiff final product. These can be filled with reinforcing materials that reduce tendency to brittle fracture, many of which are much less of a concern than glass fibre or carbon fibre but still increase stiffness. Precast to near form,and finish machine is a good way to go. Even for a one off, it can be economical to cut a mold in something like beaver-barf, cast a blank with appropriate reinforcement, then finish machine. I don’t have enough experience here to make any real recommendations, as I tend to use these for their fill and bonding properties.
If you are looking for off-the-shelf, you might find it worth your while to get a sample kit that has a selection of materials you are interested in to do test pieces.