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We investigate the stellar kinematics of the bulge and disk components in 826 galaxies with a wide range of morphology from the Sydney-AAO Multi-object Integral-field spectroscopy (SAMI) Galaxy Survey. The spatially-resolved rotation velocity (V) and velocity dispersion ($σ$) of bulge and disk components have been simultaneously estimated using the penalized pixel fitting (pPXF) method with photometrically defined weights for the two components. We introduce a new subroutine of pPXF for dealing with degeneracy in the solutions. We show that the V and $σ$ distributions in each galaxy can be reconstructed using the kinematics and weights of the bulge and disk components. The combination of two distinct components provides a consistent description of the major kinematic features of galaxies over a wide range of morphological types. We present Tully-Fisher and Faber-Jackson relations showing that the galaxy stellar mass scales with both V and $σ$ for both components of all galaxy types. We find a tight Faber-Jackson relation even for the disk component. We show that the bulge and disk components are kinematically distinct: (1) the two components show scaling relations with similar slopes, but different intercepts; (2) the spin parameter $λ_R$ indicates bulges are pressure-dominated systems and disks are supported by rotation; (3) the bulge and disk components have, respectively, low and high values in intrinsic ellipticity. Our findings suggest that the relative contributions of the two components explain, at least to first order, the complex kinematic behaviour of galaxies.