There's a rare type of diatonic accordion which is now being marketed by a company which imports fancy items made in the former Soviet Bloc countries. It has 25 buttons in two rows on the treble side, and another 25 in three rows for the bass. Apparently it's quite popular in Russia.
On this instrument, each button plays a single note, the same for both the push and the pull.
If you can read Russian, there is a good link on Squeezebox 101 (see links) which gives instructions on playing this instrument.
The right hand side is simple in layout but rather convoluted for playing. The notes alternate between rows, and each octave is exactly the opposite of its adjacent octaves.
The layout shown in the instructions is for a C instrument. The leftmost key of the inner row appears to be a chromatic helper key and is not part of the scale. Starting with the second key from the left in the inner row the note sequence is d-f-a-c-e-g-b-d-f-a-c. In the outer row, the two leftmost keys are not part of the scale. Starting with the third key, the outer row sequence is c-e-g-b-d-f-a-c-e-g-b.
The left side has single low bass notes on the inner row. The sequence is a-g-f-e-d-c-b-a-f#. The middle and inner rows sequences of low single notes and related chords at higher pitches. The middle-row sequence is d-d-a-a-e-e-b-b. The outer row sequence is f-f-c-c-g-g-d-d.
I have not had my hands on one of these accordions, so these notes here are rather sketchy. This is from the P. Londonov self-teaching course of more than 90 chapters. The whole thing is imaged on the web, so anyone who reads Russian can gain a good education on this instrument.
The only American firm I know of that imports this type of instrument is The Sovietski Collection which deals in fancy gift and novelty items. They sometimes have these instruments in their catalog.