Symbolism, Imagery, Allegory
Notice how the plebs are always trying to "rise" up (1.1.47) against the ruling class of aristocrats? That's one example of the play's vertical imagery, which all has to do with power relations--who's on "top.".
Remember, Roman society is hierarchical. The powerful patricians are at the top of the food chain and the plebs are at the very bottom. So, when the plebeians challenge the ruling class, they're threatening to flatten or level the entire society. This is why Cominius warns "That is the way to lay the city flat; [...] In heaps and piles of ruin" (3.1.256-259). Cominius isn't just worried that a civil war will literally destroy the city--he's worried that the social hierarchy will crumble as well. Shakespeare emphasizes this threat when the plebs threaten to throw Coriolanus down from the Tarpeian Rock for being an enemy to the people (3.1).
Movin' on up
The play's vertical imagery also relates to Coriolanus' dramatic "rise" and "fall" as the play's tragic hero. As an aristocrat and a war hero, Coriolanus has definitely risen to the top of Roman society. But then he kneels down before his domineering mom after returning home from war (2.1). You could definitely argue that his kneeling down marks the beginning of his downfall.
One of the final images in the play is of Tullus Aufidius standing on top of Coriolanus' dead body. According to one of the most famous stage directions in all of Shakespeare, Coriolanus "falls" after being stabbed and "Aufidius stands on him" (5.6.156.Stage Directions).
And you know what? We're not surprised. It's clear that Aufidius wants to show once and for all his dominance over his arch enemy. It's also clear that Shakespeare wants us to know that Coriolanus has just experienced a major downfall—just like all of his other tragic heroes. There's a reason they don't call it "falling up."