An algebra A over a field K is said to be invertible if it has a basis B consisting only of units; if B−1 is again a basis, A is invertible-2, or I2. The question of when an invertible algebra is necessarily I2 arises naturally. The study of these algebras was recently initiated by López-Permouth, Moore, Szabo, Pilewski [13], [14]. In this paper, we prove several positive results on this problem, answering also some questions and generalizing a few results from these papers. We show that every field is an I2 algebra over any subfield, and that any subalgebra of the rational functions field K(x) which strictly contains K[x], with K an algebraically closed field, has a symmetric basis B=B−1. Using this, we expand the class of examples of algebras known to be invertible or I2 with several classes, such as semiprimary rings over fields K≠F2 satisfying some additional mild condition. We also show that every commutative affine invertible algebra is almost I2 in the sense that it becomes I2 after localization at a single element.