When people say something revolves around something, but “doesn’t spin”, this is what they mean:
1
._.
O
2
O ._.
3
O
._.
4
._. O
The moon is tidally locked with Earth, however, so it’s more like this:
1
._.
O
2
O |:
3
O
.-.
4
:| O
The moon spins relative to space. We can see that because we see how the face changes orientation.
However, relative to Earth, we always see one side. For a little observer dude on Earth, they would not see the moon spin.
Einstein famously said time is relative. He also said speed is relative. Everyone thinks they’re not moving relative to themselves, that everything else is moving. We think we are still, on Earth, when in reality, our planet is hurtling at 100,000 kilometres per hour around our Sun, which is circling the galaxy at mind-bogglingly fast speed.
Spinning is like speed. The Earth is spinning a lot, but we don’t notice it. We see the sun spin around the Earth from the Earth, because we are on the Earth. This is why people thought everything revolved around the Earth for a long time, and it was understandable why they would think that.
In the same way, the Moon spinning around the Earth is relative. Someone on Earth would not see it. Someone in space would.
I have only been talking about the Earth/Moon relationship, when the Earth revolves around the Sun, which revolves around the supermassive black hole at the center of the galaxy, further convoluting the “spinning” of the Moon.
If we took everything else out of the picture, and left the Moon going in seemingly very random movement in space, you would clearly see it is spinning. It is the other stuff which makes it hard to answer this question.
tl;dr Saying the moon doesn’t spin is almost like saying the Sun revolves around the Earth, but less dumb and more justified.
Final answer: Yes, the moon spins.