The following diagrams are covering spaces of the wedge sum of two circles $S^1 \vee S^1$.
In Hatcher's Algebraic Topology, Hatcher has this short remark on page 71.
Note that in $(7)$ the group of deck transformations is $\mathbb{Z}_4$ while in $(8)$ it is $\mathbb{Z}_2 \times \mathbb{Z}_2$.
The definition for a deck transformation we use is that similar to Munkre's definition in Topology.
Definition: Given a covering map $p:E \to B$, a deck transformation is a homeomorphism $h: E \to E$ such that $p = p \circ h$.
For $(7)$, each deck transformation can be view geometrically as a rotation of $0,\pi/2,\pi,$ or $3\pi/2$. This explains why the group of deck transformations is $\mathbb{Z}_4$. In $(8)$, two of the deck transformations are a rotation of $0$ and a rotation of $\pi$. However, it is unclear to me what the other two transformations are. In my lecture notes, my professor noted that the deck transformations for $(8)$ can be viewed as rotations and reflections. However, by reflecting $(8)$ at all, the orientation of each of the segments reverses. Doesn't this indicate that $p \neq p \circ h$ for a reflection homeomorphism $h$?
My question: What are the deck transformations geometrically for $(8)$ and $(9)$ and how do you compute them? Furthermore, how do you argue that there are no other deck transformations?