We propose a novel recurrent variational netwo8=k]irk, SegMorph, to perform concurrent segmentation and motion estimation on cardiac cine magnetic resonance image (CMR) sequences. Our model establishes a recurrent latent space that captures spatiotemporal features from cine-MRI sequences for multitask inference and synthesis. The proposed model follows a recurrent variational auto-encoder framework and adopts a learnt prior from the temporal inputs. We utilise a multi-branch decoder to handle bi-ventricular segmentation and motion estimation simultaneously. In addition to the spatiotemporal features from the latent space, motion estimation enriches the supervision of sequential segmentation tasks by providing pseudo-ground truth. On the other hand, the segmentation branch helps with motion estimation by predicting deformation vector fields (DVFs) based on anatomical information. Experimental results demonstrate that the proposed method performs better than state-of-the-art approaches qualitatively and quantitatively for both segmentation and motion estimation tasks. We achieved an 81% average Dice Similarity Coefficient (DSC) and a less than 3.5 mm average Hausdorff distance on segmentation. Meanwhile, we achieved a motion estimation Dice Similarity Coefficient of over 79%, with approximately 0.14% of pixels displaying a negative Jacobian determinant in the estimated DVFs.