Consider this Rubik's slide. With these moves (and their inverses): $$\text{Vertical shift}\: v=(147)(258)(369)$$$$\text{Rotation}\: c=(12369874)$$$$\text{Horizontal shift}\: h=(123)(456)(789)$$ Also consider the $5$ square to be the origin, the $(258)$ row to be the $Y$-axis and the $(456)$ row to be the $X$-axis
Question:
I'm looking for combinations of $v,h,c$ that flip the Rubik's slide about it's $X$-axis and $Y$-axis
Goal:
I'm trying to show that I can transpose any two squares of the Rubik's slide without changing any of the other $7$ squares. So If I can show all of the possible transpositions of neighbors in the Rubik's slide. I can prove a more encompassing theorem using the fact that a transposition of two squares is just the product of transpositions of neighbors. I'm hoping to use the answer to the above question to greatly decrease the amount of cases I'll have to deal with.
I know I've asked a similar question before, but I tried my best to make this question as succinct as possible where my previous question had multiple questions strewn throughout it. I also tried to reword it to make it more understandable
In your previous question you quote a constructive proof showing how any two squares in the grid can be transposed.
Use that to swap first 1 and 7, then 2 and 8, then 3 and 9.