I am trying to find symmetries of square pyramid to color its faces. We have $4$ rotational symmetries such as identity, $90^\circ$ , $180^\circ$, $270^\circ$. The rotational symmetries are clear and sensible. However, when I come to $4$ reflectional symmetries such as with respect to midpoint of two opposite edges and with respect to diagonal of the base square, I confused. The reason is that when I apply the reflectional symmetry, then the square pyramid will not be seen the same, because its direction will reserve. What I meant by "its direction will reserve" is that its top point look at above in the original position, but look at bottom when reflectional symmetry applied.
I confused because as far as I know when we apply reflection or rotation, we must obtain the same appearance. Am I wrong in that? If not, how will we obtain its reflections.
Please be clear in your answer or comment , because I just started learning this topic
MORE EXPLANATION: When I looked at comments, I saw that I could not explain what I meant, so let think a classical square based pyramid and called its lateral faces as $1,2,3,4$ such that
Now , lets write its symmetries :
$90^\circ :(1432)$
$180^\circ :(13)(42)$
$270^\circ :(1234)$
Reflection $1$ by midpoints of opposite sides (Think the red line as plane dividing the pyramid into to pieces): $(1)(3)(24)$.
Look at the picture:
However, when we reflect it like that, we obtain such a appearance:
Reflection $2$ by midpoints of opposite sides : $(13)(2)(4)$
Reflection by diagonal : $(32)(14)$
Reflection by diagonal : $(34)(12)$
Let's think about "Reflection $1$ by midpoints of opposite sides". When you look at its picture, you will see that its apex pointing out the bottom side while it was pointing out "the sky or above" before reflection. I said that I'm confused, because this pyramid:
is not the same as the original one:
However, we must have obtained same appearance. I could not manage to reflect it when the apex pointing out above.