The rocking response of embedded foundations in sand is studied, combining centrifuge modelling and numerical analysis. In total, 51 centrifuge model tests are conducted at ETH Zurich, varying the embedment ratio D/B and the factor of safety against vertical loading (bearing capacity), F_s. The experimental results are used to validate three-dimensional finite-element models, which are subsequently employed for a parametric study. The initial two phases of non-linear response are studied, namely quasi-elastic and non-linear response, while the third phase (plastic response) will be examined in a forthcoming publication. Regarding small-strain rocking stiffness, K_r (first phase), an extended formula is proposed, accounting for soil inhomogeneity along the embedded sidewalls. K_r is shown to decrease with F_s, due to initial soil yielding from vertical loading. The formula is further extended to account for the role of F_s, offering improved estimation of small-strain rocking stiffness. With respect to the non-linear response (second phase), it is shown that the degradation curves of rocking stiffness with rotation can become approximately dimensionless, if the rotation θ is normalised to a characteristic rotation, θ_s. The increase of D/B leads to a reduction of the normalised rotation θ/θ_s at which the non-linear response is initiated, due to the increasing participation of sidewall friction.