Not all triangles are the same - Even if they look alike !

It is already hinted at in the first tutorial. The tutorial contains an unsolvabe puzzle which is only there to demonstrate the moves-UI & how to switch quickly between puzzles.
Advanced players will notice & understand why it is unsolvable :
Depending on the starting position, the playground symmetries, and the rotators used some triangles cannot be in a specific location with a specific rotation (alignment)

A group of triangles consists of those triangles that are interchangeable. 
This is obvious when triangles only have a color.
When they do have faces and direction markers too, some can indeed look exactly identical, but they can NOT be exchanged !