The centrosome is a non-membrane-bound cellular compartment consisting of 2 centrioles surrounded by a protein coat termed the pericentriolar material (PCM). Centrioles generally remain physically associated together (a phenomenon called centrosome cohesion), yet how this occurs in the absence of a bounding lipid membrane is unclear. One model posits that pericentriolar fibres formed from rootletin protein directly link centrioles, yet little is known about the structure, biophysical properties, or assembly kinetics of such fibres. Here, I combine live-cell imaging of endogenously tagged rootletin with cell fusion and find previously unrecognised plasticity in centrosome cohesion. Rootletin forms large, diffusionally stable bifurcating fibres, which amass slowly on mature centrioles over many hours from anaphase. Nascent centrioles (procentrioles), in contrast, do not form roots and must be licensed to do so through polo-like kinase 1 (PLK1) activity. Transient separation of roots accompanies centriolar repositioning during the interphase, suggesting that centrioles organize as independent units, each containing discrete roots. Indeed, forced induction of duplicate centriole pairs allows independent reshuffling of individual centrioles between the pairs. Therefore collectively, these findings suggest that progressively nucleated polymers mediate the dynamic association of centrioles as either 1 or 2 interphase centrosomes, with implications for the understanding of how non-membrane-bound organelles self-organise.