Accurate characterization of visual attributes such as spiculation, lobulation, and calcification of lung nodules in computed tomography (CT) images is critical in cancer management. The characterization of these attributes is often subjective, which may lead to high inter- and intra-observer variability. Furthermore, lung nodules are often heterogeneous in the cross-sectional image slices of a 3D volume. Current state-of-the-art methods that score multiple attributes rely on deep learning-based multi-task learning (MTL) schemes. These methods, however, extract shared visual features across attributes and then examine each attribute without explicitly leveraging their inherent intercorrelations. Furthermore, current methods treat each slice with equal importance without considering their relevance or heterogeneity, which limits performance. In this study, we address these challenges with a new convolutional neural network (CNN)-based MTL model that incorporates multiple attention-based learning modules to simultaneously score 9 visual attributes of lung nodules in CT image volumes. Our model processes entire nodule volumes of arbitrary depth and uses a slice attention module to filter out irrelevant slices. We also introduce cross-attribute and attribute specialization attention modules that learn an optimal amalgamation of meaningful representations to leverage relationships between attributes. We demonstrate that our model outperforms previous state-of-the-art methods at scoring attributes using the well-known public LIDC-IDRI dataset of pulmonary nodules from over 1,000 patients. Our model also performs competitively when repurposed for benign-malignant classification. Our attention modules provide easy-to-interpret weights that offer insights into the predictions of the model.