Single-walled carbon nanotubes (SWCNTs) have substantial application potential in many branches of technology. Considering that the spatial order of carbon atoms in SWCNTs determines their properties, the characteristics of a material can be adjusted for a specific application by using only selected types of SWCNTs. Unfortunately, SWCNT purification is challenging as the mechanisms of the currently available sorting techniques are not fully understood. The aqueous two-phase extraction (ATPE) method is a promising approach in this context as it offers SWCNT separation in an aqueous environment, so elucidation of its workings is essential. To come closer to achieving this goal, we synthesized a new range of surfactants based on ionic liquids equipped with amphiphilic carbohydrate moieties. We then examined how the modification of their structure influences SWCNT partitioning in two-phase systems composed of poly(ethylene glycol) and dextran. We concluded that the capacity of the ATPE method for purification of SWCNTs depended upon establishing a sufficient degree of order or disorder in the aqueous medium caused by the introduction of chemical compounds of a kosmotropic and chaotropic nature, respectively.