AC Adapter Polarity 230816

Hello ETO forum,

The context is a twelve volt circuit that uses a PIR sensor to trigger a microcontroller, ATtiny85,
to turn on 12 volt LEDs to come on for a minute.
The circuit is powered by a 12 volt, two amp AC/DC adapter, a wall wart. The adapter is small enough (3"x1.5"x 1") that the light fixture was designed to hold the adapter. So the distance between the adapter and the PCB barrel jack is about an inch.
The wire between the adapter and the barrel jack on the PCB, as it was purchased, is several feet long. So the adapter wire was shortened so the wire fits better with light fixture.
The adapter wire used in the fixture shown had a red wire and a white wire inside a black sleeve
so it was obvious which wire went to which. But on the adapter wire being shortened today there is no difference between the two wires. They are just two black wires.
It is realized that manufacturers can do what they want. But most of the manufacturers of the small adapters have evolved to using 2.5 x 5.5 x 11mm center positive plugs, for instance.

Will the two conductors on the adapter have polarity?
If so what will happen if the polarity is reversed?
Is there a method, using a multimeter, to test for the correct hookup?

Thanks.

Allen Pitts