  We define even very stable Higgs bundles and study the Hitchin map restricted
to their upward flows. In the GL(n) case we classify the type (1,...,1)
examples, and find that they are governed by a root system formed by the roots
of even height. We discuss how the spectrum of equivariant cohomology of
quaternionic Grassmannians, 4n-spheres and the real Cayley plane appear to
describe the Hitchin map on even cominuscule upward flows. The even upward
flows in question are the same as upward flows in Higgs bundle moduli spaces
for quasi-split inner real forms. The latter spaces have been pioneered by
Oscar Garc\'ia-Prada and his collaborators.
