Bipartite 3-regular counting problems with mixed signs

We prove a complexity dichotomy for a class of counting problems expressible as bipartite 3-regular Holant problems. These are also counting CSP problems where every constraint has arity 3 and every variable is read-thrice. For every problem of the form $\mathrm{Holant}\left(f\right|{=}_3)$, where f is any integer (or equivalently, rational)-valued ternary symmetric constraint function on Boolean variables, we prove that it is either P-time computable or #P-hard, depending on an explicit criterion of f. The constraint function can take both positive and negative values, allowing for cancellations. In addition, we discover a new phenomenon: there is a set $F$ with the property that for every $f\in F$ the problem $\mathrm{Holant}\left(f\right|{=}_3)$ is planar P-time computable but #P-hard in general, yet its planar tractability is by a combination of a holographic transformation by $\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right]$ to FKT together with an independent global argument.