You see a reflection because glossy surfaces aren't optimum. If you look in the DIY screen section, you will see that flat white paint is generally used.
If you look at any professional screen material, you will see that it doesn't resemble a mirror at all. It more resembles a fine white (or silver, or gray) sandpaper (although the grains are usually not made of little rocks but of tiny glass beads)
You have a somewhat darkened theater, so I would, if I were you, look into white screens, starting with the budget brands. If you have a dedicated theater, you probably can go with a fixed screen.
Or just paint your glossy board with flat white paint.