Twitter is one of the most widely used social networks globally but is also notorious for harbouring significant levels of hate speech. This paradox has garnered the attention of companies and governments concerned about how digital hate can fragment communities and incite real-world violence. While extensive research has focused on detecting hate speech, there is a lack of comprehensive analysis of the actors involved, their characteristics, and their interactions within the online environment. Common assumptions categorise users into merely two groups—haters and non-haters—overlooking the existence of other groups that may more accurately represent the dynamics of hate dissemination. Additionally, existing user classification models often rely on large volumes of tweet data, a limitation given the restricted access to the Twitter API. Social networks are also frequently visualised using graphs where edges represent only superficial relationships. This study addresses these gaps through five research questions. We propose formal user clustering methods and develop a classifier that uses exclusively profile attributes—information more readily obtainable from the network. We also introduce a more nuanced definition of interaction for graph edges, based on ideological support or opposition between users. To conduct our analysis, we have extracted two complementary datasets: (i) a keyword-based corpus of 3.3M Spanish tweets containing hate-associated terms, of which 1.6M unique tweets were retained after filtering and (ii) a user-based corpus comprising timelines of $\approx$3,000 users linked to hate speech, totalling over 3 M tweets. Our results reveal the existence of three primary user classes—haters, upstanders, and neutrals—in contrast to the conventional binary classification. We demonstrate that profile attributes are reliable indicators for automatically classifying users and find significant statistical differences between these classes. Finally, we develop a graph visualisation tool to assist authorities in analysing interactions among different user types, providing a useful exploratory tool to support the analysis of online hate and inform potential mitigation strategies.