Pooling is a critical operation in convolutional neural networks for increasing receptive fields and improving robustness to input variations. Most existing pooling operations downsample the feature maps, which is a lossy process. Moreover, they are not invertible: upsampling a downscaled feature map can not recover the lost information in the downsampling. By adopting the philosophy of the classical Lifting Scheme from signal processing, we propose LiftPool for bidirectional pooling layers, including LiftDownPool and LiftUpPool. LiftDownPool decomposes a feature map into various downsized sub-bands, each of which contains information with different frequencies. As the pooling function in LiftDownPool is perfectly invertible, by performing LiftDownPool backward, a corresponding up-pooling layer LiftUpPool is able to generate a refined upsampled feature map using the detail subbands, which is useful for image-to-image translation challenges. Experiments show the proposed methods achieve better results on image classification and semantic segmentation, using various backbones. Moreover, LiftDownPool offers better robustness to input corruptions and perturbations.