Centrioles have a unique, conserved architecture formed by three linked triplet microtubules arranged in nine-fold symmetry. The mechanisms by which these triplet microtubules are formed are not understood, but likely involve the noncanonical tubulins delta-tubulin and epsilon-tubulin. Previously, we found that human cells deficient in delta-tubulin or epsilon-tubulin form abnormal centrioles, characterized by an absence of triplet microtubules, lack of central core protein POC5, and a futile cycle of centriole formation and disintegration (Wang et al., 2017). Here, we show that human cells lacking either of the associated proteins TEDC1 and TEDC2 have these same phenotypes. Using ultrastructure expansion microscopy, we identified the roles of these proteins and triplet microtubules in centriole architecture by mapping the locations of centriolar proteins throughout the cell cycle. We find that mutant centrioles have normal architecture during S-phase. By G2-phase, mutant centrioles grow to the same length as control centrioles, but fail to recruit inner scaffold proteins of the central core. Instead, the inner lumen of centrioles is filled with an expanded proximal region, indicating that these proteins, or the triplet microtubules themselves, may be required for recruiting central core proteins and restricting the length of the proximal end. During mitosis, the mutant centrioles elongate further before fragmenting and disintegrating. All four proteins physically interact and TEDC1 and TEDC2 are capable of interacting in the absence of the tubulins. These results support an AlphaFold Multimer structural prediction model for the tetrameric complex, in which delta-tubulin and epsilon-tubulin are predicted to form a heterodimer. TEDC1 and TEDC2 localize to centrosomes and are mutually dependent on each other and on delta-tubulin and epsilon-tubulin for localization. These results indicate that delta-tubulin, epsilon-tubulin, TEDC1, and TEDC2 function together in promoting robust centriole architecture. This work also lays the groundwork for future dissection of this complex, which will provide a basis for determining the mechanisms that underlie the assembly and interplay between compound microtubules and inner centriole structure.