We show that a unique Jacobi matrix can be reconstructed from two types of mixed data: 1. its two eigenpairs, 2. its eigenvalues and the eigenvalues of its submatrix obtained by removing the first two rows and columns and its 1-1 entry. For the second case, the condition of existence is provided. In addition, we summarize the equivalent sets of parameters used to recover Jacobi matrices and show some direct connections.