  We study the Carrollian limit of the (general) quadratic gravity in four
dimensions. We find that in order for the Carrollian theory to be a
modification of the Carrollian limit of general relativity, the parameters in
the action must depend on the speed of light in a specific way. By focusing on
the leading and the next-to-leading orders in the Carrollian expansion, we show
that there are four such non equivalent Carrollian theories. Imposing
conditions to remove tachyons (from the linearized theory), we end up with a
classification of Carrollian theories according to the leading-order and
next-to-leading-order actions. All modify the Carrollian limit of general
relativity with quartic terms of the extrinsic curvature. To the leading order,
we show that two theories are equivalent to general relativity, one to $R+R^2$
theory, and one to the general quadratic gravity. To the next-to-leading order,
two are equivalent to $R+R^2$ while the other two to the general quadratic
gravity. We study the two theories that are equivalent to $R+R^2$ to the
leading order and write their magnetic limit actions.
