We study those fully irreducible outer automorphisms ϕ \phi of a finite rank free group F r F_r which are parageometric, meaning that the attracting fixed point of  ϕ \phi in the boundary of outer space is a geometric R \mathbf {R} -tree with respect to the action of  F r F_r , but  ϕ \phi itself is not a geometric outer automorphism in that it is not represented by a homeomorphism of a surface. Our main result shows that the expansion factor of ϕ \phi is strictly larger than the expansion factor of ϕ − 1 \phi ^{-1} . As corollaries (proved independently by Guirardel), the inverse of a parageometric outer automorphism is neither geometric nor parageometric, and a fully irreducible outer automorphism ϕ \phi is geometric if and only if its attracting and repelling fixed points in the boundary of outer space are geometric R \mathbf {R} -trees.