Scientists tested eighteen reindeer with light, and found that their eyes responded to both visible light and ultraviolet. Unlike other mammals, their cornea lets ultraviolet light pass into the eye. They don't have specialized rod and cone cells in their retina to detect ultraviolet like some animals, but their normal cells respond to low intensity stimulation.
Even though most birds can see ultraviolet light, not many mammals do. Some rodents and bats are an exception.
Reindeer are a type of caribou highly adapted for life in the northern tundra. They have special noses with convoluted turbinate bones for warming cold air. They also have hooves that grow spongy pads in summer to navigate mud, and shrink down to expose the hard hoof in winter to walk on ice and snow. It's not surprising that they have special eyesight too.
Why would they need to see ultraviolet light?
Like other caribou, they live on lichens, leaves, sedges and grasses. It just so happens that lichens, their major food source in winter, absorb ultraviolet light.
That would make them appear black amid all the background snow that reflects ultraviolet, and be easier to find. Urine also absorbs ultraviolet. Their vision may enable them to detect when predators have been marking their territories.