Prior ion-mobility mass-spectrometry (IMS-MS) studies of polyethylene glycol (PEG) ions have identified only two out of many sharply different observed structures: Linear shapes with several individually solvated singly charged cations at high charge states z (beads on a string), and single multiply charged globules at low z. The present study is devoted to assign all other existing structures of PEG ions, for the first time reaching masses of 100 kDa and charge states up to z = 10. There are at most z different structures at charge state z. All involve a single globule carrying n charges, tied to one or several appendices bearing z - n separate charges in a beads-on-a-string configuration. All sharp shape transitions observed at decreasing ion mass involve ejection of one elementary charge (sometimes two) from the shrinking globule into the growing linear appendage. This picture is supported by molecular dynamics simulations and approximate calculations of electrical mobilities for computed structures.