These pianos were made by several piano manufacturers. I have come across 3 companies so far that manufactured the transposing piano; the Heintzman Co. of Canada, Weser Bros. and the Sohmer Co. both of New York.
Irving Berlin, who was a self-taught pianist and played only on the black keys, had several models of this instrument to transpose the music into different keys.
There is a mechanism below the keyboard that allows the key bed to shift laterally left or right; this allows the player to play the same keys on the keyboard as previous while having the music played in a different key.
This is a rare Heintzman & Co. Transposing upright piano, which used to be produced some time back. This instrument has 98 keys and songs can be played in 10 different keys …11 in fact, including the one you are in. In 35 yrs. of service to the industry I have only ever seen two of these. Here are some photos of this model of transposing piano.
Here is a thread from Piano World Tech forum where I posted this as a topic. What ended up happening in the thread was an informal count of these instruments across Canada. So far I have come across more that 30 of these rare transposing pianos. Enjoy!