Currently, I am a bit confused by two representations of the dihedral group. In my search for an answer, I came across this unanswered question, which explains these two representation far more beautiful than I can. Here, it is suggested that there actually is no isomorphism between these two representations. But how can that be the case when they describe the same symmetry?
Or does there exists an isomorphism between these two representations? And if so what would this isomorphism be? (My attempts fail due to the fact that after flipping the rotations will be in different directions in the two representations.)
I think the question that you reference is somewhat unclear. I think the confusion is between (1) describing a symmetry $\theta$ of a regular polygon by describing a sequence of simple symmetries (rotations and reflections) that result in $\theta$, as opposed (2) to describing it by numbering the vertices in its initial position and then giving the numbering of the vertices in each position after applying the symmetry $\theta$. (I have a recollection of a blog by Terence Tao that compared these two approaches, but unfortunately I can't locate it just now.)
In approach (1), it is clear how to get the description of two symmetries given the descriptions of the two symmetries - you just concatenate the sequences of simple symmetries. In approach (2), it is not. To see why approach (2) is actually equivalent to approach (1) and to see how to get the approach (2) descriptions of the composition of two symmetries described using approach (2), you can first condense the approach (2) descriptions by imagining that you were initially standing at vertex 1 facing clockwise. An approach (2) description is then determined by a statement of the form "I am now at vertex $i$ facing clockwise" or "I am now at vertex $i$ facing anticlockwise". In this simplified version of approach (2), it should be clear that you get the descriptions of the composition of the symmetries by taking the difference of the vertex positions and (in a suitable sense) of the directions. Each of the approach (1) descriptions has an equivalent approach (2) description and vice versa, so the two methods of description are equivalent.