How to tell $i$ from $-i$?

Simple: there are two distinct values whose square is $-1$, so you choose one of them and call it $i$. The other one, then, is $-i$. It doesn't matter which of them you choose, just as it doesn't matter which direction in space you call $x$.

If you construct $\Bbb C$ as $\Bbb R[X]/(X^2+1)$ the roots are indistinguishable. You just choose one and identify it with the point $(0,1)$ in the Gauss-Argand plane. That chosen one will be $i$.

If, on the other hand you construct $\Bbb C$ as the set of pairs $(a,b)\in\Bbb R^2$ with suitable addition and multiplication, then $\pm i=(0,\pm1)$.

The two constructions are of course isomorphic but not canonically isomorphic, due to the arbitrarity of the choice of a root which is the effect of existence of a trivial $\Bbb R$-automorphism, namely complex conjugation.