Looks like Mahogany, most likely an Asian variety.

Satinwood of a particular cut.
Casey

Thanks guys. It would be easier if you'd agree! :)

Using Google images you can look at veneers until you go blind and see many photos that would be contenders. Mahogany and satinwood are two of the most likely.
In reality there is no certainty; the person who veneered it has probably been dead for 40 years.
There are ways of examining a specimen with a microscope, but they are destructive, and costly.
We have done our best.
Casey

And...
Even if you were to determine the species with certainty, what is the purpose?

If you just want to call it the right name, that is a good enough reason.

But if you're trying to match the piece with some new wood, forget it!