They sure do, Ursula. I have a 'staff' of about 6 Cuban anoles inside my cage that survive very nicely on the bugs on my orchids. Wish I could say they get all the bugs but of course they don't. I have the yellow sticky trap cards hanging around my bloomers to attract thrips and there are always lots of those. They're small enough to come through the screening, of course.
I grow my Brugmansias right outside the cage, so I'm hoping they are attracting all the mites away from the orchids most of the time. Lots of different strategies at work when you grow orchids outdoors here.