  We describe explicitly the algebra of Spin(9)-invariant,
translation-invariant, continuous valuations on the octonionic plane. Namely,
we present a basis in terms of invariant differential forms and determine the
Bernig-Fu convolution on this space. The main technical ingredient we introduce
is an extension of the invariant theory of the Lie group Spin(7) to the
isotropy representation of the action of Spin(9) on the 15-dimensional sphere,
reflecting the underlying octonionic structure. As an application, we compute
the principal kinematic formula on the octonionic plane and express in our
basis certain Spin(9)-invariant valuations introduced previously by Alesker.
