We revisit the connection between relativistic orbital precession, the Laplace-Runge-Lenz
symmetry, and the t-channel discontinuity of scattering amplitudes. Applying this to
scalar-tensor theories of gravity, we compute the conservative potential and orbital precession
induced by both conformal/disformal-type couplings at second Post-Minkowskian order (𝒪(GN
2)), complementing the known third/first order Post-Newtonian results. There
is a particular tuning of the conformal coupling for which the precession vanishes at leading PN
order, and we show that this coincides with the emergence of a Laplace-Runge-Lenz symmetry and a
corresponding soft behaviour of the amplitude. While a single scalar field inevitably breaks this
symmetry at higher PN orders, certain supersymmetric extensions have recently been shown to have
an exact Laplace-Runge-Lenz symmetry and therefore classical orbits do not precess at any PN
order. This symmetry can be used to relate scattering amplitudes at different loop orders, and we
show how this may be used to bootstrap the (classically relevant part of the) three-loop 2 → 2
scattering of charged black holes in 𝒩 = 8 supergravity from existing two-loop
calculations.