You can make spline points symmetric across a centerline with the symmetry relation, which is very handy.
However you can still edit an individual spline point's direction and magnitude, and this will not come over to its symmetric point. I know you can "reset all handles" and that returns the direction and magnitude to their original orientation but it's a little unfortunate that I can not use those tools if I want a symmetric curve built over the centerline.
Yeah, I agree, this stinks. You'd think SW had exhausted all the sketching enhancements that could possibly be made with as few enhancements as it has seen in the past several years.
The only way I have found to do this is to make two splines, with the endpoints on the line of symmetry, one spline mirrored from the other. This does give you a split at the line of symmetry, but it also gives you symmetry that includes the spline handles.
From there I would do a fit spline of the 2 splines assuming that the spline is perpendicular to the symmetry plane.
(I love how this forum thinks "spline" is spelled wrong)
A quick thought about this is that you can dimension both the direction and magnitude of the points and create linked dimensions to impose symmetry. Extra work I know, but its at least a way to do it.