Is it true that
$f(t)\star g(-t)$
is not the same as
$f(t)\star h(t)$
if $h(t)=g(-t)$?
($\star$ means cross-correlation).
The example I was thinking of was if $g(t)=t$. Then $(f(t)\star g(-t))(t)=\int_{-\infty}^\infty f(\tau)(-t+\tau)d\tau$, but $(f(t)\star h(t))(t)=\int_{-\infty}^\infty f(\tau)(-(t+\tau))d\tau=\int_{-\infty}^\infty f(\tau)(-t-\tau)d\tau$.
If it's true then wouldn't something like $t^2\star -t$ be ambiguous?
I think I'm most likely making a mistake here, so any help would be greatly appreciated.
Something like $t^2\star -t$ is understandable. Since you didn't define $\star$, I'm not sure if or where you made a mistake.
The $f(t)*g(-t)$ is convenient notation and is very common in engineering texts. It's convenient, because it's easier to write $f(t)*g(-t)$ than to write this with more conventional function notation: $$(f*h)(t),$$ where $h(t) = g(-t)$, and be forced to define a new function $h$.
The downside to convenience is that it mixes together the function's "name" and "argument", and can lead to awkward expressions like $(f(t)*g(-t))(t)$ if you want to evaluate this at some point $t$.