At the moment I'm somewhat confused by the concept of colored filters; common sense states that they allow only their color of light to pass through(i.e. red filter lets red light through), but, if they appear to be a specific color, wouldn't that indicate that they reflect that color?
You're right, an ideal filter can either absorb or reflect the spectrum it should not pass, and thus for that ideal filter we should not see that pass-band color reflected from the filter. Unfortunately, practical materials are less efficient and typically reflect a little of the spectrum they are supposed to pass (the degree of inefficiency depends on the type and spectrum of filter). The reason you often see a filter that has a reflection color the same as the color it is suppose to pass, is that it absorbs all other colors (so even the little that gets reflected from some internal structures quickly gets absorbed before it makes it out of the material) but the little pass-band color that does reflect from internal structures does make it out of the material for us to see, because that color isn't absorbed much. On top of that, the eye-brain system causes us to consider something clearly colored (we say its "red") even though it may barely have more red than other colors.
I would expect that the highest quality filters do not reflect any of their colour (red for example) and only appear red because everything behind them appears red. As it blocks out all the other wavelength's of visible light. A red filter like that would appear black in a completely blue room, however it is not actually completely black, the same would apply to a blue filter in a red room.
This would be different for filters that are likely to be used in a school classroom where the light filters do indeed reflect a small portion of their respective light and a red filter is genuinely red (as it appears).
Hope this helps!:-)