Using carbon dioxide (CO₂) to make recyclable thermoplastics could reduce greenhouse gas emissions associated with polymer manufacturing. CO₂/cyclic epoxide ring-opening copolymerization (ROCOP) allows for >30 wt % of the polycarbonate to derive from CO₂; so far, the field has largely focused on oligocarbonates. In contrast, efficient catalysts for high molar mass polycarbonates are underinvestigated, and the resulting thermoplastic structure-property relationships, processing, and recycling need to be elucidated. This work describes a new organometallic Mg(II)Co(II) catalyst that combines high productivity, low loading tolerance, and the highest polymerization control to yield polycarbonates with number average molecular weight (M_n) values from 4 to 130 kg mol^-1, with narrow, monomodal distributions. It is used in the ROCOP of CO₂ with bicyclic epoxides to produce a series of samples, each with M_n > 100 kg mol^-1, of poly(cyclohexene carbonate) (PCHC), poly(vinyl-cyclohexene carbonate) (PvCHC), poly(ethyl-cyclohexene carbonate) (PeCHC, by hydrogenation of PvCHC), and poly(cyclopentene carbonate) (PCPC). All these materials are amorphous thermoplastics, with high glass transition temperatures (85 < T_g < 126 °C, by differential scanning calorimetry) and high thermal stability (T_d > 260 °C). The cyclic ring substituents mediate the materials' chain entanglements, viscosity, and glass transition temperatures. Specifically, PCPC was found to have 10× lower entanglement molecular weight (M_e)_n and 100× lower zero-shear viscosity compared to those of PCHC, showing potential as a future thermoplastic. All these high molecular weight polymers are fully recyclable, either by reprocessing or by using the Mg(II)Co(II) catalyst for highly selective depolymerizations to epoxides and CO₂. PCPC shows the fastest depolymerization rates, achieving an activity of 2500 h^-1 and >99% selectivity for cyclopentene oxide and CO₂.