So I'll be honest, I never use tangent planes. It's just a personal thing with inventor that outside of a sketch, you'll never find me telling anything to be tangent. It always seems to cause more problems than it solves (in the long run).
All that to just say I don't have a ton of experience with them and can't say for sure that there's no way to do what you're wanting to do as it pertains to strictly working with tangent planes.
There are, however, many other approaches to achieve a similar result. Especially if this is in a part file. And I think you could probably redefine your current plane to use a different definition method and maintain your current results. I cannot, however, speak for what may happen when you try to switch from east to west. That all depends on the remaining features and how many references there are to the feature you wish to change. That can be very touchy and can sometimes require a careful hand if you intend to have something change on the fly.
There are two ways I'd suggest approaching this personally:
A) Create a plane that rotates around the cylindrical axis by whatever angle (reliable in part files, less so in assemblies). Then offset a plane from that by the radius of the cylinder (best to actually use the dimension that defines the feature rather than just inputting a number just in case.. something changes). The offset plane may just be the redefinition of the plane you're currently using.
B) Assuming you have a sketch with a circle somewhere that defines this cylindrical face (or that you can create one that references it), this may look like two circles concentric on a plane that are used to extrude a pipe for example. You can create 2 construction lines from the center point out to the edge: one should be constrained to the origin axis (vertical or horizontal) and the other free to rotate. Then create an angular dimension between the two. The plane definition comes from selecting the rotating line and then it's endpoint on the edge of the cylinder.
Both of these methods should create a plane that will stay oriented properly with respect to the cylinder and allow you to move the plane and accompanying features to suit your needs.
This is just what I would do though. It's a little extra work compared to telling a plane to be tangent to a surface but I find a little more time for a more robust and/or flexible solution always pays off in the end.
Hope this helps!