Pic(C), the group of divisor classes on C, has two distinguished elements, 0 and K, hence two natural involutions: E—> - E, and E—>K-E, where K is the canonical divisor class. The first involution leaves invariant the subgroup Pic(0) = the Jacobian variety of classes of degree zero, while the second leaves invariant the coset Pic(g-1). The second involution also leaves invariant (by Riemann - Roch) the image W(g-1) of the symmetric product Sym^(g-1)(C) in Pic(g-1). Both these involutions are algebraic.
Since a smooth plane quartic is a non - hyperelliptic curve of genus 3, on which a canonical divisor is cut by any line L in the plane, C^(2) imbeds isomorphically onto W(2) in Pic(2), and your construction is an example of the second involution; in particular it is algebraic. I.e. your involution is the restriction of E—> K-E to W(2) ≈ C^(2).
As Will Sawin and Sasha pointed out, if D is any divisor class of degree g-1, effective or not, then translation by D takes the second involution on Pic(g-1) to some involution on Pic(0), and it takes it to the first involution, E—> - E, if and only if 2D = K, if and only if D is a fixed point of the second involution, E—>K-E.
Indeed Riemann associated to any canonical homology basis of C, a “theta function” on Pic(0), i.e. an even function whose zero locus, the “theta divisor”, is thus invariant under the first involution E—> - E. Riemann’s famous theorem says that such a homology basis determines also a specific divisor class D, with 2D = K, called a “theta characteristic”, which translates W(g-1) isomorphically onto the theta divisor, carrying the involution E —> K-E to the involution E—> -E.
Since Pic is a complete variety, and the map E—>2E on Pic(C) has finite fibers, there always exist divisor classes D with 2D = K, in fact exactly 2^(2g) of them, and (I believe) Riemann showed that 2^(g-1).(2^g -1) of them are effective. In your case this says 28 effective divisors D exist with 2D = K = L, where D = the pair of points of contact of one of the 28 bitangents.