Reverse, transpose, invert can all mean to change to the contrary or opposite side or position.
Reverse is the most general of these terms, implying a change to the opposite not only in side or position but also in direction, order, sequence, relation, or bearing; thus, to reverse a coin is to turn it upside down; to reverse a process is to follow the opposite order of sequence; to reverse a judgment is to change a previous judgment to another that is contrary to it; to reverse a garment or part of a garment is to turn it inside out; to reverse the direction of a locomotive is to make it go backward instead of forward.
Transpose implies a change in position, usually by reversing the order of two or more units (as letters or words) or by an exchange of position. But transpose often, especially in grammar or anatomy, implies merely a change in the natural order or position.
Invert implies a change from one side to another chiefly by turning upside down but occasionally, especially in surgery, by turning inside out or outside in. In its secondary senses it approaches reverse but applies within narrower limits <invert the order of words in a sentence> <invert the relation of cause and effect> <the custom … to invert now and then the order of the class so as to make the highest and lowest boys change places —Thomas Moore >