No puzzle for you, and probably not for me, but there is one for him.
Remember when you were what, 20, 12, 5 years old and you first looked at one of these circuits?
There is no reason to think it's not properly wired, but a drawing of the circuit should still help him. Why do people often post urls with circuit drawings? To help.
A DPDT switch is two SPDT switches that are connected mechanically (so one handle operates both) but not electrically. A DPDT can be converted to a 4-way switch by cross connecting the non-common terminals.
If you call the terminals of a 3-way A,B,C (where C is common) and for a DPDT add 1 or 2 for which switch, then connecting 1A and 2B to one terminal, and 1B and 2A to another will make it a 4-way switch.
It is a DPDT but the only terminals are on the switched contact and one way the switch is made, (straight through). The other side it is made is cross connected to the straight through side, in effect swapping the leads but the terminal is not available..
So it's a DPDT switch made into a crossover kind of switch. That sound about right?
Yup
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