  The Kolakoski sequence $S$ is the unique element of $\left\lbrace 1,2
\right\rbrace^{\omega}$ starting with 1 and coinciding with its own run length
encoding. We use the parity of the lengths of particular subclasses of initial
words of $S$ as a unifying tool to address the links between the main open
questions - recurrence, mirror/reversal invariance and asymptotic density of
digits. In particular we prove that recurrence implies reversal invariance, and
give sufficient conditions which would imply that the density of 1s is
$\frac{1}{2}$.
