  We provide several constructions in differential KO-theory. First, we
construct a differential refinement of the $\hat{A}$-genus and a pushforward
leading to a Riemann-Roch theorem. We set up a differential refinement of the
Atiyah-Hirzebruch spectral sequence (AHSS) for differential KO-theory and
explicitly identify the differentials, including ones which mix geometric and
topological data. We highlight the power of these explicit identifications by
providing a characterization of forms in the image of the Pontrjagin character.
Along the way, we fill gaps in the literature where K-theory is usually worked
out leaving KO-theory essentially untouched. We also illustrate with examples
and applications, including higher tangential structures, Adams operations, and
a differential Wu formula.
