The normal equations in the analysis of variance with suitable side conditions give a unique set of estimates, $\tilde\beta_i$, of the parameters $\beta_i$. These estimates are unique linear forms in the actual observations. In the solutions to the normal equations they appear as linear forms in the treatment and block totals; these totals are not independent and so the forms in them are not unique. Thus the normal equations, while giving a unique solution vector, admit an infinite number of pseudo-inverses of the matrix $X' X$. In this paper the relationship between the two most common pseudo-inverses is discussed.
"Pseudo-Inverses in the Analysis of Variance." Ann. Math. Statist. 35 (2) 895 - 896, June, 1964. https://doi.org/10.1214/aoms/1177703596