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According to Animal Wellness Magazine, head pressing is one of the biggest signs that something could be terribly wrong with your fur baby. If your pet does this, it could indicate damage to its nervous system. This could be caused by things like toxic poisoning, brain tumors, or even a stroke.
So if you see your pet pressing its head up against the wall (or any other object for that matter), don't assume they are just playing around. Don't brush this behavior off. Don't attempt to treat them at home. Take them to the vet to rule out any serious illnesses, and seek treatment in a worst case scenario. Even if it's nothing, it's better to be safe than sorry.