  We classify a natural collection of GL(2,R)-invariant subvarieties, which
includes loci of double covers, the orbits of the Eierlegende-Wollmilchsau,
Ornithorynque, and Matheus-Yoccoz surfaces, and loci appearing naturally in the
study of the complex geometry of Teichmuller space. This classification is the
key input in subsequent work of the authors that classifies "high rank"
invariant subvarieties, and in subsequent work of the first author that
classifies certain invariant subvarieties with "Lyapunov spectrum as degenerate
as possible". We also derive applications to the complex geometry of
Teichmuller space and construct new examples, which negatively resolve two
questions of Mirzakhani and Wright and illustrate previously unobserved
phenomena for the finite blocking problem.
