What are the secret hats and room items?

Besides the hats and room items that open up with your streak, every tab but Window and Sign has secret ones. Until you find one, it shows only as a dark shadow with a question mark and ??? for its name, and tapping it says only that it's a surprise. Each secret opens up on its own milestone as you keep going in the ministry and care for your pet; which one is part of the surprise. None depend on hours, none need a buddy, and none can be missed for good, so you can find any of them later. The first time you see one after it opens up, it says New; then pick it like any other item. Secrets you've found stay open, sync with your pet like other items, and come back from a backup. If a secret you expected hasn't shown up, open your pet's details again after your records sync; secrets are worked out on each device from what's stored there.