I have a doubt in Chapter 5, section 2 (Group Motions of the Plane) of Michael Artin. Page 159 has a paragraph which describes how computations in group $M$ of all rigid motions in a plane (translation, rotation, and reflection) can be done using symbols as given in the attached picture. While it is easy to understand the compositions ticked in green, I have not understood the compositions boxed in red. References 2.3, 2.4 from the book also has been added to the picture.
Would be grateful for any help in understanding these.