Reconstruction of High Dynamic Range (HDR) from Low Dynamic Range (LDR) images is an important computer vision task. There is a significant amount of research utilizing both conventional non-learning methods and modern data-driven approaches, focusing on using both single-exposed and multi-exposed LDR for HDR image reconstruction. However, most current state-of-the-art methods require high-quality paired {LDR;HDR} datasets with limited literature use of unpaired datasets, that is, methods that learn the LDR-HDR mapping between domains. This paper proposes CycleHDR, a method that integrates self-supervision into a modified semantic- and cycle-consistent adversarial architecture that utilizes unpaired LDR and HDR datasets for training. Our method introduces novel artifact- and exposure-aware generators to address visual artifact removal. It also puts forward an encoder and loss to address semantic consistency, another under-explored topic. CycleHDR is the first to use semantic and contextual awareness for the LDR-HDR reconstruction task in a self-supervised setup. The method achieves state-of-the-art performance across several benchmark datasets and reconstructs high-quality HDR images. The official website of this work is available at: https://github.com/HrishavBakulBarua/Cycle-HDR