For American sensibilities born from classical liberal values, you're correct.

The political "left" is a big umbrella term, like the "right" is. Generally speaking, the left broadly wants distribution of resources and management to be more horizontal. The debate is how to achieve that. Some want total worker control, others believe a democratically elected government should be empowered to distribute wealth equitably. The most extreme and dystopian form is a parental centralised authority enforcing equality should do it, as you mentioned.

On the right, they may believe in personal freedom but still uphold that human society still thrives in hierarchy. "You're homeless? It is your fault and not my problem." Many on the right believe you have to work hard to achieve your goals but through your own blood, sweat and tears. You have to keep up or you will be out competed. Some on the right still believe that government still has some role to help others, while some disagree. The most extreme form on the right is fascism, which advocates for total control of everything and order and hierarchy are utmost importance.