We study a sample of 28 S0 galaxies extracted from the integral-field spectroscopic (IFS) survey CALIFA. We combine an accurate two-dimensional (2D) multi-component photometric decomposition with the IFS kinematic properties of their bulges to understand their formation scenario. Our final sample is representative of S0s with high stellar masses ($M_{star}/M_{sun} > 10^{10}$). They lay mainly on the red sequence and live in relatively isolated environments similar to that of the field and loose groups. We use our 2D photometric decomposition to define the size and photometric properties of the bulges, as well as their location within the galaxies. We perform mock spectroscopic simulations mimicking our observed galaxies to quantify the impact of the underlying disc on our bulge kinematic measurements ($lambda$ and $v/sigma$). We compare our bulge corrected kinematic measurements with the results from Schwarzschild dynamical modelling. The good agreement confirms the robustness of our results and allows us to use bulge reprojected values of $lambda$ and $v/sigma$. We find that the photometric ($n$ and $B/T$) and kinematic ($v/sigma$ and $lambda$) properties of our field S0 bulges are not correlated. We demonstrate that this morpho-kinematic decoupling is intrinsic to the bulges and it is not due to projection effects. We conclude that photometric diagnostics to separate different types of bulges (disc-like vs classical) might not be useful for S0 galaxies. The morpho-kinematics properties of S0 bulges derived in this paper suggest that they are mainly formed by dissipation processes happening at high redshift, but dedicated high-resolution simulations are necessary to better identify their origin.
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