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Current best practices recommend using textual descriptions to make data visualizations accessible to blind and low vision (BLV) screen reader users. While recent research has explored laying such descriptions out hierarchically to enable reading varying levels of detail, the textual descriptions remain fixed: their syntax and semantics are set by the visualization author or tool, and cannot be changed by a BLV user based on their preferences or task-specific needs. In this paper, we explore four characteristics of customizations for hierarchical textual descriptions of visualizations: presence, or what content is present in the description; verbosity, or the length and conciseness of the content; ordering, or the sequencing of content; and, duration, or how long a particular customization lasts. We instantiate these methods as extensions to Olli, an open source library that converts web-based visualizations into hierarchical textual structures, and evaluate our work through a mixed-methods study with 13 BLV participants. Users reported that customization is crucial to their agency and that being able to change the four characteristics helps them efficiently carry out their desired tasks on the data. However, differences in preferred defaults, prior experiences, and enthusiasm for customization indicate that there is no one-size-fits-all system even for customization itself: both accessible data visualizations and user interfaces for customizing them must be flexible enough to meet a variety of needs.