A planar tangle is a planar closed simple smooth curve constructed from quarter circle arcs. These curves represent planar configurations of a fidget toy called a Tangle; which is constructed from connecting freely rotating quarter circular tubes. If you imagine that each pair of connected arcs are allowed to rotate in 3-space around the axis of connection, then there are two natural moves to create new planar tangles from known tangles. These moves induce a adjacency graphs on classes of move-equivalent tangles. This presentation develops properties such as maximum degree and bipartitionability of these graphs and develops an algorithm for generating the adjacency graph of a given class of move-equivalent tangles.