Yes, centrioles reproduce, or more accurately, they duplicate.
Centriole Duplication
The provided reference clarifies the process of centriole duplication:
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Timing: Concurrent with DNA replication, centrioles also begin to duplicate.
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Mechanism: Each existing centriole (the "mother" centriole) gives rise to a new "daughter" centriole. The daughter centriole forms at a right angle to the outer wall of the mother centriole.
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Development: By metaphase, the new daughter centriole has developed both a cartwheel structure and an outer wall.
Therefore, centrioles don't simply divide; they act as templates for the formation of new centrioles. This process ensures that each daughter cell receives the correct number of centrioles during cell division.