Image demoiréing aims to remove structured moiré artifacts in recaptured imagery, where degradations are highly frequency-dependent and vary across scales and directions. While recent deep networks achieve high-quality restoration, their full-precision designs remain costly for deployment. Binarization offers an extreme compression regime by quantizing both activations and weights to 1-bit. Yet, it has been rarely studied for demoiréing and performs poorly when naively applied. In this work, we propose BinaryDemoire, a binarized demoiréing framework that explicitly accommodates the frequency structure of moiré degradations. First, we introduce a moiré-aware binary gate (MABG) that extracts lightweight frequency descriptors together with activation statistics. It predicts channel-wise gating coefficients to condition the aggregation of binary convolution responses. Second, we design a shuffle-grouped residual adapter (SGRA) that performs structured sparse shortcut alignment. It further integrates interleaved mixing to promote information exchange across different channel partitions. Extensive experiments on four benchmarks demonstrate that the proposed BinaryDemoire surpasses current binarization methods. Code: https://github.com/zhengchen1999/BinaryDemoire.