Is this what you progress to after mastering a "Tutor Model" Concertina?

Look Here

Funny, I looked at that last night and thought "Hang on a bit, that's two instruments bolted together".

I bet the tone quality between the ends is nice and even

While it does seem possible that the instrument was constructed from parts of what were originally two instruments (call that scenario 0) , other possibilities suggest themselves:

1) The buttons on one end were damaged together at some point, and the person doing the repairs had only the other kind for replacement. (I would guess that the metal-topped buttons were the later addition.)

2) Like 1, but somebody had disassembled the one end, had it apart for too long, and lost the original buttons. (May the cleaning service didn't recognize them for what they were and discarded them?)

For 2, other bits -- such as the bottom bushings -- would probably also be of a different stock. For 1, they might have been reused.

3) Non-symmetrical instruments were occasionally produced, as the Wheatstone ledgers list a few with one wood and one metal end. While it may be easier to imagine a reason for that than for different buttons, the original purchaser may have ordered it that way for personal -- or even medical -- reasons. I loaned Chris Timson my G/D Jeffries because after his mild stroke he found it easier to control than his own under his left hand. Yet the only discernable difference between the two instruments was a tiny difference in button size.

0) Returning to the conjecture of two original instruments being later combined, one could ask whether internal parts are stamped with more than one serial number. If parts on both ends are all stamped the same, then that should elilminate this scenario.

RETRACTION: While it was fun speculating as to alternate reasons for the difference(s), an examination of the pictures on eBay shows that the end with the metal buttons has inlaid brass roses and the end with the bone buttons doesn't. Combining parts from what were originally two different instruments -- even two instruments of significantly different age -- seems the only likely origin for that beast. And that does raise questions of whether the reed pans and reeds were also from two different instruments and, if so, whether the sound quality of the two ends might differ significantly.