There are some pieces whose middle part is shaped like a Technic axle, but that have elements on both ends that make it impossible to insert them into a normal axle or Technic hole:
This makes me wonder whether the axle shape here actually allows for a connection or was only chosen for stability, production reasons, or similar. With other words, is there any piece (or combination of pieces) that connects a Technic axle “sideways”, i.e., without going over the ends?
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.
The minifigure spanner can be attached sideways to an axle (as well as a pin and stud):
This doesn’t hold in all directions, however.
This part (Technic, Link 1 x 16 #2637) appears to be an axle with two connectors on each end. Bricklink classifies this piece as a Technic Link and not an axle because it doesn't perform most of the same functions as an axle.
I believe the decision to use the Technic 'axle' shape for the length of the piece rather than a round piece, is because it functions more like an axle than a pin. Pins connect bricks together and are two to three bricks long. Axles can be much longer, one of which is the same length as this piece.
Of the 23 sets this part appears in there aren't any sets in which bricks have been attached to the axle part of the piece, nor does there to be any obvious way to do so.
