The synchrosqueezing method aims at decomposing 1D functions into superpositions of a small number of “Intrinsic Modes”, supposed to be well separated both in time and frequency. Based on the unidimensional wavelet transform and its reconstruction properties, the synchrosqueezing transform provides a powerful representation of multicomponent signals in the time–frequency plane, together with a reconstruction of each mode. In this paper, a bidimensional version of the synchrosqueezing transform is defined, by considering a well-adapted extension of the concept of analytic signal to images: the monogenic signal. We introduce the concept of “Intrinsic Monogenic Mode”, that is the bidimensional counterpart of the notion of Intrinsic Mode. We also investigate the properties of its associated Monogenic Wavelet Decomposition. This leads to a natural bivariate extension of the Synchrosqueezed Wavelet Transform, for decomposing and processing multicomponent images. Numerical tests validate the effectiveness of the method on synthetic and real images.