Under multi-headed dependency grammar, a parse is a connected DAG rather than a tree. Such formalisms can be more syntactically and semantically expressive. However, it is hard to train, test, or improve multi-headed parsers because few multi-headed corpora exist, particularly for the projective or planar case. To help fill this gap, we observe that link grammar already produces undirected planar graphs, and so we wanted to determine whether these could be converted into directionalized dependency parses. We use Integer Linear Programming to assign consistent directions to the labeled links in a corpus of several thousand parses produced by the Link Grammar Parser, which has broad-coverage hand-written grammars of English as well as Russian and other languages. We find that such directions can indeed be consistently assigned in a way that yields valid multi-headed dependency parses. The resulting parses in English appear reasonably linguistically plausible, though differing in style from CoNLL-style parses of the same sentences; we discuss the differences.