Both a perfect mirror and a perfectly white sheet of paper reflect all the light that hits them. The difference is that the paper scatters the light so what reaches your eye is a mixture of all the light hitting the paper, while the mirror reflects the light without scattering. If you're interested in details the reflection from a mirror is specular while the reflection from the piece of paper is diffuse.
If you swap the sheet of paper for a mirror the light isn't scrambled. If you're looking at e.g. a beachball some of the cone cells in your eye will receive light from the red bits of the beachball, some from the yellow sand, some from the blue sky and so on. So the individual cone cells are seeing the colour of the scene you're reflecting in the mirror not the colour of the mirror itself. In this sense the mirror doesn't have a colour. However if you average over all the cone cells, i.e. mix all the light up, you'll get the same colour as when you had the sheet of paper there. That's why I claim the mirror has the same colour as the piece of paper.
Of course you could argue the paper doen't really have a colour either. After all the paper may look white in daylight, but it would look green in a jungle or red in a tomato sauce factory.
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