Production of exclusive dijets in diffractive deep inelastic $e^pm p$ scattering has been measured with the ZEUS detector at HERA using an integrated luminosity of 372 pb$^{-1}$. The measurement was performed for $gamma^*-p$ centre-of-mass energies in the range $90 < W < 250$ GeV and for photon virtualities $Q^2 > 25$ GeV$^2$. Energy and transverse-energy flows around the jet axis are presented. The cross section is presented as a function of $beta$ and $phi$, where $beta=x/x_{rm I!P}$, $x$ is the Bjorken variable and $x_{rm I!P}$ is the proton fractional longitudinal momentum loss. The angle $phi$ is defined by the $gamma^*-$dijet plane and the $gamma^*-e^pm$ plane in the rest frame of the diffractive final state. The $phi$ cross section is measured in bins of $beta$. The results are compared to predictions from models based on different assumptions about the nature of the diffractive exchange.
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