  Given two orthogonal projections $\{P,Q\}$ in a non commutative tracial
probability space, we prove relations between the moments of $P+Q$, of
$\sqrt{-1}(PQ-QP)$ and of $P+QPQ$ and those of the angle operator $PQP$. Our
proofs are purely algebraic and enumerative and does not assume $P,Q$
satisfying Voiculescu's freeness property or being in general position. As far
as the sum and the commutator are concerned, the obtained relations follow from
binomial-type formulas satisfied by the orthogonal symmetries associated to $P$
and $Q$ together with the trace property. In this respect, they extend those
corresponding to the cases where one of the two projections is rotated by a
free Haar unitary operator or more generally by a free unitary Brownian motion.
As to the operator $P+QPQ$, we derive autonomous recurrence relations for the
coefficients (double sequence) of the expansion of its moments as linear
combinations of those of $PQP$ and determine explicitly few of them. These
relations are obtained after a careful analysis of the structure of words in
the alphabet $\{P, QPQ\}$. We close the paper by exploring the connection of
our previous results to the so-called Kato's dual pair. Doing so leads to new
identities satisfied by their moments.
