I am trying to work out if the compass swing platform is part of the runway or taxiway.
It could be on a taxiway, but it can also be outside the movement area (e.g. on the apron or ramp in the US). As far as I know, it is never on the runway.
There is a compass rose on a taxiway at Petaluma Municipal Airport (O69). You can see it on the following image (from Google Maps). It is painted in front of the hold short line of runway 11.
Most airports will however have a dedicated position for the compass rose. This image (taken from this question) shows an example at Ramona Airport (KRNM):
It is not part of the taxiway as indicated by the yellow line, which separates the non-movement area from the movement area.