  A normal pair of H\"older triangles is the union of two normally embedded
H\"older triangles satisfying some natural conditions on the tangency orders of
their boundary arcs. It is a special case of a surface germ, a germ at the
origin of a two-dimensional closed semialgebraic (or, more general, definable
in a polynomially bounded o-minimal structure) subset of $R^n$. Classification
of normal pairs considered in this paper is a step towards outer Lipschitz
classification of definable surface germs. In the paper \cite{BG} we introduced
a combinatorial invariant of the outer Lipschitz equivalence class of normal
pairs, called $\sigma\tau$-pizza, and conjectured that it is complete: two
normal pairs of H\"older triangles with the same $\sigma\tau$-pizzas are outer
Lipschitz equivalent. In this paper we prove that conjecture and define
realizability conditions for the $\sigma\tau$-pizza invariant. Moreover, only
one of the two pizzas in the $\sigma\tau$-pizza invariant, together with some
admissible permutations related to $\sigma$ and $\tau$, is sufficient for the
existence and uniqueness, up to outer Lipschitz equivalence, of a normal pair
of H\"older triangles.
