In this paper, several inverse Kalman filtering problems are addressed, where unknown parameters and/or inputs in a filtering model are reconstructed from observations of the posterior estimates that can be noisy or incomplete. In particular, duality in inverse filtering and inverse optimal control is studied. It is shown that identifiability and solvability of the inverse Kalman filtering is closely related to that of an inverse linear quadratic regulator (LQR). Covariance matrices of model uncertainties are estimated by solving a well-posed inverse LQR problem. Identifiability of the considered inverse filtering models is established and least squares estimators are designed to be statistically consistent. In addition, algorithms are proposed to reconstruct the unknown sensor parameters as well as raw sensor measurements. Effectiveness and efficiency of the proposed methods are illustrated by numerical simulations.