Between homeomorphism type and Tukey type
Call a compact space $X$ pin homogeneous if every two points $a,b$ are pin equivalent, meaning that there exists a compact space $Y$, a quotient map $f\colon Y\to X$, and a homeomorphism $g\colon Y\to Y$ such that $gf^{-1}\{a\}=f^{-1}\{b\}$. We will prove a representation theorem for pin equivalence; transitivity of pin equivalence will be a corollary. Pin homogeneity is strictly weaker than homogeneity and pin equivalence is strictly stronger than Tukey equivalence. Just as with topological homogeneity, no infinite compact $F$-space is pin homogeneous. On the other hand, $X\times 2^{\chi(X)}$ is pin homogeneous for every compact $X$. And there is a compact pin homogeneous space with points of different $\pi$-character.