Pole Bean
Old charts keep pole beans and beets apart, and no mechanism has ever been shown for the conflict. The practical caution is simpler: a bean trellis throws long shade over a crop that wants full sun, so keep any trellis on the north side of the beet row.
Field Mustard
Field mustard grows fast and broad and often shades out beet seedlings while drawing down the same water and nitrogen. It also feeds flea beetles, which then move onto beet greens. Keep it as a separate cover-crop patch, turned in well before beets are sown.
Swiss Chard
Chard and beet are the same species and host the same troubles — leaf spot, downy mildew, and leaf miners move freely between them. Growing them side by side stacks the risk and undoes any rotation you keep elsewhere.
Good King Henry
Beet and chard share this crop's family, its leaf miner, and its leaf spot and mildew problems. Keep beet rows well away from the perennial bed so the standing crown does not act as a year-round reservoir for those pests and diseases.
Quinoa
Beets sit in the same plant family as quinoa and host the same leaf miners and Cercospora leaf spot, so growing them side by side stacks both problems. Keep them in separate beds and respect a two-year gap between family members.