To understand the physics behind a homopolar motor, first and foremost you have to understand that besides the electric current in a current-carrying wire there is also a magnetic current (which the contemporary science calls magnetic field). The two currents are inseparably bound to each other very similar to the water swirling current and the cavitation spiral current which an underwater propeller creates. The water swirling current corresponds to the electric current, whereas the cavitation spiral current corresponds to the magnetic current. Watch the cavitation spiral current in this short video:

https://youtu.be/Y7k7p1RirkI?t=11

Both the electric and the magnetic current flow in the same direction from the positive to the negative terminal of a battery through the connecting wire, just as the water current and the cavitation current flow in the same direction. The flow of the electric current is counter-clockwise, while the flow of the magnetic current is clockwise.

Please pay a particular attention to this: the magnetic spiral HAS A DIRECTION OF MOTION. It is clockwise from the positive to the negative terminal of the battery.
Thus, the Plus of the spiral is directed towards the minus terminal of the battery.

For better understanding, please look at this spiral and imagine that you are turning it to the right holding it on the near end (picture below):

In this way you can make a simulation of the magnetic spiral motion in a current-carrying wire.

Now, what is a homopolar motor? In this video (3:22 on the timeline) is presented its simplest version to carry out.

https://youtu.be/KyGpd28wg_4?t=204

Now, look please at the diagram below:

If the minus-pole of the magnet is facing downwards (figure a), then the magnet together with the screw is turning clockwise (when viewed from below). Why? Because the minus-pole of the magnet is attracting the spiral magnetic current; thus the magnet begins to turn in accordance with the motion of the spiral magnetic current.

If the plus-pole of the magnet is facing downwards (figure b), then the magnet together with the screw is turning counter-clockwise. Why? Because the plus-pole of the magnet is repelling the spiral magnetic current; thus the magnet begins to turn contrary to the motion of the spiral magnetic current.

To better understand what I am talking about, you have to see also these answers:

What is an electric current?

Is positive and negative electricity nomenclature arbitrary?

What is a battery?

Are electromagnetic waves transverse or longitudinal?