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.
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.