Let $C$ be a smooth plane quartic defined over a field $K$. Denote by $J$ its Jacobian, and by $C^{(2)}$ its symmetric square. Since $C$ is a smooth plane quartic, it is non-hyperelliptic, and hence the morphism $C^{(2)}\longrightarrow J$.

The symmetric square of $C$ has an involution defined as follows: let $(P,Q)\in C^{(2)}(K)$, let $L_{P,Q}$ be the line connecting $P$ and $Q$ in $\mathbb{P}^2_K$, and the tangent line at $P$ if $P=Q$. The scheme theoretic intersection of $L_{P,Q}$ with $C$ has multiplicity $4$, and hence there is a degree $2$ effective divisor $(S,T)\in C^{(2)}(K)$ such that $(L) = (P) + (Q) + (S) + (T)$. We define the involution $(P,Q)\mapsto (S,T)\in C^{(2)}(K)$.

My questions are:

1. Is the involution $C^{(2)}\longrightarrow C^{(2)}$, defined above, algebraic?

2. The Jacobian $J$ has a canonical involution coming from multiplication by $(-1)$. Under what circumstances does the involution defined above coincide with multiplication by $(-1)$. I know for example this would be the case if the curve $C$ has a flex, but is it necessary?