Let be a, not necessarily closed, linear relation in a Hilbert space $\sH$ with a multivalued part $\mul A$. An operator in $\sH$ with $\ran B\perp\mul A^{**}$ is said to be an operator part of when $A=B \hplus (\{0\}\times \mul A)$, where the sum is componentwise (i.e. span of the graphs). This decomposition provides a counterpart and an extension for the notion of closability of (unbounded) operators to the setting of linear relations. Existence and uniqueness criteria for the existence of an operator part are established via the so-called canonical decomposition of . In addition, conditions are developed for the decomposition to be orthogonal (components defined in orthogonal subspaces of the underlying space). Such orthogonal decompositions are shown to be valid for several classes of relations. The relation is said to have a Cartesian decomposition if $A=U+\I V$, where and are symmetric relations and the sum is operatorwise. The connection between a Cartesian decomposition of and the real and imaginary parts of is investigated.