I want to use a piece of fabric as an inexpensive reflector.
Is there any way to find out if it is pure white, so that I can be sure it will not add a colour cast to the light it reflects?
There are a few things you could do:
The first method will probably be a lot more reliable as it won't succumb to lighting colour shifts or the vagaries of Raw processing. However, provided it's not massively off (in which case you'd be able to see that clearly) it probably doesn't matter if it's pure white.
Colour temp is relative so the only time it matters is if you're using it alongside other light modifiers which are pure white, so my suggestion, if this is the case, would be to try the first option and compare it to your other modifiers either by eye, or better by shooting images of them under controlled lighting.
IMO, it's mostly a waste of time to even try. It's color is going to shift over time, and if it's like most things that are "white", it'll include some "brighteners" -- fluorescent dyes, so the ultraviolet content in the light makes it glow a bit. These mean the color of the fabric varies based in the light (specifically its UV content).
Instead, therefore, put your reflector under the intended light source, and use that to set your white balance. If you're shooting film, the equivalent is to take a shot of the reflector material under the intended light, about like you normally would of a grey card. When you print, adjust for that to be neutral, then use the same color balance for the remaining shots on that roll.
The only way is to photograph the fabric reflecting the intended light source onto a typical scene you want to shoot. This is because the color of something depends on the frequency spectrum of the electromagnetic radiation shining on it (including invisible frequencies such as near ultraviolet--think "black lights" and fluorescence) and the differential ways in which that reflector absorbs, scatters, and even polarizes the different color frequencies. In particular, you don't necessarily need a "perfect" white: you only need a reflector that re-emits the incoming light. If the ambient light does not include certain portions of the spectrum--and this is true of almost all lights--then your reflector does not need to reflect those proportions, either. So requiring a "pure white" reflector is both unattainable and unnecessary.
Checking is a simple enough operation: photograph a neutral card with just the light and then illuminated as much as possible by the reflector (using the same light source). Use the same white balance settings for the photos. Visually compare the two images, or--for a more scientific approach--quantitatively compare their RGB histograms. There will be a change in color, however slight, but you can decide whether it is important for the photography you intend to do.