This paper studies questions about duality between crossings and non-crossings in graph drawings via the notions of thickness and antithickness. The "thickness" of a graph $G$ is the minimum integer $k$ such that in some drawing of $G$, the edges can be partitioned into $k$ noncrossing subgraphs. The "antithickness" of a graph $G$ is the minimum integer $k$ such that in some drawing of $G$, the edges can be partitioned into $k$ thrackles, where a "thrackle" is a set of edges, each pair of which intersect exactly once. (Here edges with a common endvertex $v$ are considered to intersect at $v$.) So thickness is a measure of how close a graph is to being planar, whereas antithickness is a measure of how close a graph is to being a thrackle. This paper explores the relationship between the thickness and antithickness of a graph, under various graph drawing models, with an emphasis on extremal questions.