Gas-phase autoxidation - the sequential regeneration of peroxy radicals (RO2) via intramolecular hydrogen shifts (H-shifts) followed by oxygen addition - leads to the formation of organic hydroperoxides. The atmospheric fate of these peroxides remains unclear, including the potential for further H-shift chemistry. Here, we report H-shift rate coefficients for a system of RO2 with hydroperoxide functionality produced in the OH-initiated oxidation of 2-hydroperoxy-2-methylpentane. The initial RO2 formed in this chemistry are unable to undergo alpha-OOH H-shift (HOOC-H) reactions. However, these RO2 rapidly isomerize (>100 s(-1) at 296 K) by H-shift of the hydroperoxy hydrogen (ROO-H) to produce a hydroperoxy-substituted RO2 with an accessible alpha-OOH hydrogen. First order rate coefficients for the 1,5 H-shift of the alpha-OOH hydrogen are measured to be similar to 0.04 s(-1) (296 K) and similar to 0.1 s(-1) (318 K), within 50% of the rate coefficients calculated using multiconformer transition state theory. Reaction of the RO2 with NO produces alkoxy radicals which also undergo rapid isomerization via 1,6 and 1,5 H-shift of the hydroperoxy hydrogen (ROO-H) to produce RO2 with alcohol functionality. One of these hydroxy-substituted RO2 exhibits a 1,5 alpha-OH (HOC-H) H-shift, measured to be similar to 0.2 s(-1) (296 K) and similar to 0.6 s(-1) (318 K), again in agreement with the calculated rates. Thus, the rapid shift of hydroperoxy hydrogens in alkoxy and peroxy radicals enables intramolecular reactions that would otherwise be inaccessible.