I tried to investigate symmetry of two relatively simple partial differential equations. Firstly I define what I mean by symmetry: change of variables $ t \rightarrow -t $ or $ x \rightarrow -x $ or function $ u \rightarrow -u $ are my symmetries, they can combine together. My equation is:
$$ u_{t} = uu_{x} $$
Then I concluded, that this equation don't have symmetries like S(t) (change of just $t$ does not lead to the same equation), S(x), S(u), S(xtu). But it has symmetries: S(xt) (change of $x$ and $t$ leads to the same equation), S(xu), S(tu). I am especially not sure about $u \rightarrow -u$ symmetry. I just plug (-) sign for every $u$ in the equation. Do have these symmetries any name? If it is correct, do they say something about given equation? The second example is equation:
$$ u_{t} = uu_{xxx} $$
and it has actually the same symmetries as the first equation (S(xt), S(xu), S(tu)).
Yes, both of your question are true. Specifically, if $u(t,x)$ is a solution of $$ u_t=uu_x $$ then so are $v_1(t,x):=-u(-t,x)$, $v_2(t,x):=-u(t,-x)$ and $v_3(t,x):=u(-t,-x)$. The same holds for your second equation $u_t=uu_{xxx}$. Of course these simetries have implications on the dynamics of the equation. For instance, if you have an equation with all of these simetries that also have soliton solutions (traveling waves), you would immediately conclude that if $u(t,x)=\phi(x-t)$ is a solitary wave solution of your equation, then so is $v(t,x)=-\phi(x+t)$. In particular, you would conclude the existence of "antisolitons" solutions moving to the left (which is not always true, there are many equations which have no "antisolitons" nor solitons moving to the left). And the same for the other simmetries.
About the names, I am not sure, but I think I would call "space-time reversal symmetry" to the map $(t,x)\mapsto(-t,-x)$ (maybe space-time inversion?). I don't think that the other two have names.
A final comment is that usually when these PDEs have Hamiltonian structure, these symmetries have associated conservation laws (by Nöether's Theorem). Hence, apriori, the existence of these symmetries (for hamiltonian PDEs at least) restricts how bad/bizarre can be the dynamic of the solution.
Edit: Notice also that both of your equations are invariant under space-time translations. There you have another two symmetries. That means: if $u(t,x)$ is a solution, then so is $u(t+t_0,x+x_0)$ for any $t_0,x_0\in\mathbb{R}$.