Q:

x(t) and x(-t) are flipped (left to right) signals of each other

$x(t)$ and $x(-t)$ are flipped (left to right) signals of each other.  but $x(2t-3)$ and $x(-2t+3)$ do not share that kind of relation as I thought incorrectly.  Can some one comment on this?  Is there a name for the relationship of this kind of signals?  Is it important enough to study this relation and give a name?

A:

$x(t)$ and $x(-t)$ are flipped left-to-right about the point $t=0$.  that point of reflection at $t=0$ is determined by asking: "When is $t$ and $-t$ equal to the same value?"
$$ t = -t \qquad \qquad \implies t=0 $$
so it turns out that $x(2t-3)$ and $x(-2t+3)$ are also flipped left-to-right, but the point of reflection is not $t=0$.  So then, what value of $t$ is the point of reflection and how do we determine that value?  Like above, it is when the two expression of argument to the common function are equal to each other:
$$ 2t-3 = -2t+3 $$
solve for $t$ and you'll find your point of reflection.

