The study of eye gaze fixations on photographic images is an active research area. In contrast, the image sub-category of freehand sketches has not received as much attention for such studies. In this paper, we analyze the results of a free-viewing gaze fixation study conducted on 3904 freehand sketches distributed across 160 object categories. Our analysis shows that fixation sequences exhibit marked consistency within a sketch, across sketches of a category and even across suitably grouped sets of categories. This multi-level consistency is remarkable given the variability in depiction and extreme image content sparsity that characterizes hand-drawn object sketches. In this paper, we show that the multi-level consistency in the fixation data can be exploited to 1) predict a test sketch's category given only its fixation sequence and 2) build a computational model which predicts part-labels underlying fixations on objects. We hope that our findings motivate the community to deem sketch-like representations worthy of gaze-based studies vis-a-vis photographic images.