I can’t make a mirror copy of the 3D spline on the left without losing the Bezier controls. The one on the right was made with Polyplane Creator — Symmetry Creator — Edge mode — Loop Select — Polygon to Spline script.
This would usually be satisfactory, but to make a Particle Spline with an arbitrary number of evenly-spaced particles a bezier spline is required. The three symmetry tools (Creator, Modifier, Mirror) only seem to work with polygons. Forum posts about symmetry also seem to be about polygons. I tried Hiroto’s Mirror Copy Tool script but it returned error messages.