If it's hunting around level in one attitude but not the other, then it's almost certainly either a trim bias issue or a mechanical issue. I say "almost certainly" because there's a bazzilion things that
could be wrong, but those two are most likely. If it's a mechanical issue, no trim in the world will make it go away entirely, and any trim that makes it go away somewhat will be wronger than wrong.
In the shop, make absolutely sure that the controls are completely free. The surfaces should fall down from their own weight, when the airplane is level (upright or inverted), and the weight of the pushrod should pull the thing into "down" with the nose down, and "up" with the nose up. Then make sure that there is minimal slop in the controls. "Minimal" is a hard word to pin down, but if you hold the flaps and very lightly try to wiggle the elevator, you shouldn't see no free movement of the elevator at all. Up to +/- 1/16" on the trailing edge of the elevator of a Twister is probably OK, but none is better. If you're a good mechanic you know that "no friction" and "no slop" are mutually exclusive -- but you should be striving to get as close as you can.
I'm not sure at what point you'll want to cut into your airplane to fix things -- I usually figure that if I can grab the leadouts and slowly move the controls through their entire range without feeling any "sticky", and if the elevator isn't flopping around independently of the flaps, then I'll leave well enough alone.
If this isn't right, there's no point in moving on -- you could spend a lifetime (probably the plane's) trying to dodge around the problem with no luck at all.
If the controls are free, check to see if it feels neutral in both upright and inverted. If you're comfortable in upright but fighting to keep it up inverted, it'll hunt. To some extent you can fix this by biasing the handle down. In the extreme you'll get it so that it hunts upright and flies smoothly inverted. If the best you can do is to get medium-weight hunting in both upright and inverted, you may want to consider moving the CG back. Unless the plane is a real dog, you should be able to arrive at a trim arrangement that gives you effortless level flight both upright and inverted. If you're there except that there's some swooping up and down with the wind (i.e., low on one side of the circle, high on the other) then move on.
Consider taping the hinge lines. I suspect this isn't your problem, but then I don't know how advanced a pilot you are -- my "terrible hunting" may be your "almost imperceptible hunting", or visa-versa.
If I go farther, I'm just going to do a poor imitation of Paul Walker's trimming method -- so read one of Paul's many articles on the subject of trimming.
Go here. Start at the earliest article (at the bottom), work your way up to the latest.
Work is the operative word -- it's not all fun boring holes in the sky and chatting with friends -- but your plane will fly better for it. Even if you just go through his bench-trim procedure and do a half-assed job of the first few steps of the in-flight stuff you'll be miles ahead of flying an as-built plane.