Cross-modal retrieval requires building a common latent space that captures and correlates information from different data modalities, usually images and texts. Cross-modal training based on the triplet loss with hard negative mining is a state-of-the-art technique to address this problem. This paper shows that such approach is not always effective in handling intra-modal similarities. Specifically, we found that this method can lead to inconsistent similarity orderings in the latent space, where intra-modal pairs with unknown ground-truth similarity are ranked higher than cross-modal pairs representing the same concept. To address this problem, we propose two novel loss functions that leverage intra-modal similarity constraints available in a training triplet but not used by the original formulation. Additionally, this paper explores the application of this framework to unsupervised image retrieval problems, where cross-modal training can provide the supervisory signals that are otherwise missing in the absence of category labels. Up to our knowledge, we are the first to evaluate cross-modal training for intra-modal retrieval without labels. We present comprehensive experiments on MS-COCO and Flickr30k, demonstrating the advantages and limitations of the proposed methods in cross-modal and intra-modal retrieval tasks in terms of performance and novelty measures. We also conduct a case study on the ROCO dataset to assess the performance of our method on medical images and present an ablation study on one of our approaches to understanding the impact of the different components of the proposed loss function. Our code is publicly available on GitHub https://github.com/MariodotR/FullHN.git.