Is there any way to connect “sideways” to an axle?
I don't think there are any pieces LEGO intends for you to attach to the side of these axle shaped pieces. However, I'm pretty sure you could attach piece 32172 to some extent (assuming the piece being attached to is actually the right size for an axle):



It probably would not connect very well, only the clips on the sides of the socket would hold it, so it would probably slide easily, or pop of with little effort, but it could be attached.