Summary: We analyze the effect of pivotal structures (on a 2-category) on the planar algebra associated to a 1-cell as in [S. K. Ghosh, J. Algebra 339, No. 1, 27–54 (2011; Zbl 1242.46071)] and come up with the notion of perturbations of planar algebras by weights (a concept that appeared earlier in M. Burns’ thesis [“Subfactors, planar algebras and rotations ”, arXiv:1111.1362]); we establish a one-to-one correspondence between weights and pivotal structures. Using the construction of Ghosh [loc. cit.], to each bifinite bimodule over \(II_1\)-factors, we associate a bimodule planar algebra in such a way that extremality of the bimodule corresponds to sphericality of the planar algebra. As a consequence of this, we reproduce an extension of V. F. R. Jones’ theorem [arXiv e-print service, Paper No. 9909027, 122 p. (1999; Zbl 1328.46049)] (of associating ‘subfactor planar algebras’ to extremal subfactors). Conversely, given a bimodule planar algebra, we construct a bifinite bimodule whose associated bimodule planar algebra is the one which we start with, using perturbations and Jones-Walker-Shlyakhtenko-Kodiyalam-Sunder method of reconstructing an extremal subfactor from a subfactor planar algebra. The perturbation technique helps us to construct an example of a family of non-spherical planar algebras starting from a particular spherical one; we also show that this family is associated to a known family of subfactors constructed by Jones.