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 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.

Remark: I find this interesting, but the lack of upvotes may reflect the fact that this is taught in some beginning Riemann surface courses, so it may be more appropriate to math stack-exchange.

Another elementary remark is that for hyperelliptic curves, the number of effective divisors D with 2D = K seems to be the number of ways to choose them from the branch points of the canonical map, so apparently, 2^(g-1).(2^g -1) = (2g+2)choose(g-1), which was not obvious to me, but (if true) probably is to most of you.