It looks like it is a series-wound motor. That is just about the only type of motor that will run on AC or DC. The 6 brushes probably indicate that it is a 6 pole motor to increase torque but at less speed.
(Car stater motors are often 4 pole. Most other brushed motors are two pole)
I don't quite understand:-
3 brushes used by each pair of wires from the motor.
and I would have expected 3 brushes connected to one wire, 3 to another and that pair has continuity, while the other pair has no connection to the brushes.
You should wire the windings in series, with the reversing switch reversing just one of the windings.
the switch configuration is as follows.
1 2 3
4 5 6
7 8 9
There is a jumper from 1 to 8, from 2 to 7, and from 3 to 9. There is evidence of wires being hooked up to 1,2,3,4,5,6 but no wires to 7,8,9.
4 wires that test with continuity between 2 pairs.
Leaving the jumpers in place,
Line connects to 6
3 connects to one end of first pair
2 connects to other end of first pair
1 connects to neutral
4 and 5 connect to the other pair.
If the motor direction does not match the switch label, swap the wires on 4 and 5. Don't move the jumpers.
It might be worth testing it with no load on a 12V battery, with a 20 A fuse.
There is a danger of electric shock, getting fingers caught or massive overcurrent if things go wrong.