If you simply take two polarizers (the old fashond pre-autofocus kind or simple polarizing media, at least for the front one) and use an angle between them, you get the effects of polarizing filters, not just dark glasses. Am I right in assuming that a variable ND filter doesn't introduce these effects? If so, just how does it work?
No, a variable ND filter is just two polarisers, the first one being linear and the second being either linear or circular. Thus when using a variable ND you also get the effects of a polariser.
You are correct. Variable ND filters do not introduce the polarization effects that CP filters do and do not increase sky blue saturation or reduce water reflections.
They are composed of 4 layers constituting 2 circular polarizing filters. Each filter is composed of a 1/4 wave plate, which converts linear polarization to circular polarization on the outer surfaces and linear polarization layers on the inner surfaces. The 1/4 wave plate on both outer surfaces eliminates the selective attenuation of linear polarization in scenes, as well as preventing the adverse effects of linearly polarized light on an SLR's exposure sensor.
However, the X (cross effects) are still there to some degree and are worse the darker the variable ND is set and the wider the camera's aperture.
"Unfortunately" the ND filter is made of two polarisers. The simplest case is when you use two linear polarisers and depending on the angle between them, different amount of light will be removed.
Nowadays one linear filter is substituted with a circular one, due to problems with the AF system. In that case CPL filter is screwed at the top of the linear one.
Usage of polarising filters is obviously introducing problems with a "X effect" at some position.
Doug is correct — to remove any polariser effect the light is 1: circularly polarised; then 2: Two linear polarisers provide the attenuation (depending on the angle between them); then 3: a final circular polariser 'de-polarises' the light to avoid autofocus problems. If you had manual focus only and wanted polariser effects you could remove both circular polarisers leaving only the linear polarisers — but it would not be easy to align them for the desired polarisation effect AND get the attenuation you want..