I'm sure this possible in Solidworks, but for the life of me I can't figure out how to do it.

Suppose I'm modeling a part that is symmetric about say the right plane. In Sketch1 I'll add a "centerline", and then apply various symmetric relations as I constrain the sketch entities. When I create a Sketch2, which I would also like to have symmetric relations, I'd like to be able to make Sketch1 visible, and use its existing centerline, rather than adding another one in Sketch2. But as far as I can tell, if I try that, the "symmetric" relation is not available.

As a concrete example, in the image below I've selected two lines in the current sketch, and a centerline from the previous sketch. I'd like to add a symmetric relation, but all that is available is

"Colinear",

"Parallel"

"Equal".

If I duplicate the centerline in the current sketch and try again, the list expands to include

"Horizontal"

"Vertical"

"Symmetric"

"Fix"

Is there any way I can add a Symmetric relation in the first case?

If you are using SW2018 and your plane is in the center, you can mirror about that. That is the closest you are going to get to do what you want.