This paper introduces, motivates, defines, and exemplifies CROSSIDE, a design space for representing capabilities of a software for collaborative sketching in a cross-surface setting, i.e., when stakeholders are interacting with and across multiple interaction surfaces, ranging from low-end devices such as smartwatches, mobile phones to high-end devices like wall displays. By determining the greatest common denominator in terms of system properties between forty-one references, the design space is structured according to seven dimensions: user configurations, surface configurations, input interaction techniques, work methods, tangibility, and device configurations. This design space is aimed at satisfying three virtues: descriptive (i.e., the ability to systematically describe any particular work in cross-surface interaction by sketching), comparative (i.e., the ability to consistently compare two or more works belonging to this area), and generative (i.e., the ability to generate new ideas by identifying potentially interesting, under covered areas). A radar diagram graphically depicts the design space for these three virtues to enable a visual representation of one or more instances.