Crossing dependencies are known to be difficult to process in a head-initial language such as English (Levy et al., 2012). Since dependency length minimization has been claimed to be a crosslinguistic processing constraint (Futrell et al., 2015; Liu et al., 2017), the length of crossing dependencies is also expected to be minimized. This implies that the length of the intervening phrases (IL) that leads to crossing should be short. Contrary to the dependency minimization hypothesis, considerable research (e.g., Levy and Keller, 2013) has shown that in head-final languages, the head-dependent length can be long (and lead to processing facilitation). This points to processing adaptability based on certain typological properties (cf. Vasishth et al., 2011, Levy et al., 2013). We carry out a corpus study to investigate the influence of head directionality on crossing dependencies. We choose 12 languages (Head-final:6, Head-initial:6) annotated with Universal Dependencies (Nivre et al., 2016). We classify crossing dependencies as (a) Head-F: head follows the dependent, and (b) Head-P: head precedes the dependent. The results indicate that head directionality of a language has a role to play in determining both the linear and hierarchical distance in a costly structure such as crossing dependency. Crossing dependencies can have higher relative depth in head-final languages, and allow longer IL in a configuration that aligns with the default head directionality of its language. Under the assumption that corpus data reflects processing constraints, our work points to processing adaptability with respect to crossing dependencies based on certain typological properties of a language (cf. Vasishth et al., 2011; Levy et al., 2013).