According to their definition symmetrically:
Plants do not have a completely definite symmetry, but they do have a definite pattern repetition, though not all of them. (Example in favor - broccoli flowers, pine cones. Example against - red algae, bamboo)
As someone who has botany as part of his undergraduate degree, I have a few things to add and clarify:
The leaves themselves also look different from one to another (from species to species), but if we talk about technical details, all leaves must be thin, because a broad leaf is not able to do its job, which is to absorb light. You will find slightly wider leaves in leaves that are in dark places, where far red light is more common and therefore - more relative distance is required to receive the same amount of light that reaches a sunny leaf (leaves that are in direct exposure to light). In small or even prehistoric plants, they do photosynthesis in their stems instead because heat is a serious factor when it comes to light absorption - because light is a heat source and too much heat breaks down and leaves are not able to disperse heat so well compared to stems and trunks.
Therefore the first plants did not have leaves until the world cooled enough for the development of leaves.
You can also include sponges, sea lilies, sea anemones, tube worms, deep sea corals and fungi in the same category of sessile. Some of them have a defined symmetry and some are very flexible.
The immobility can be due to other reasons and not necessarily a lack of flexibility - such as food accessibility, hunting strategies, energy needs, etc.
Of course, with this comes a price they have to pay, but most of them solve it with all kinds of solutions - regeneration, stronger armor, more vigorous reproduction and even in some cases - reinventing a way to move (as in the case of sea lilies and sea anemones and both of them do not have complex muscles like us ).
Corals in general are animals that have created a symbiosis with phototrophic bacteria or algae, but they are still filter feeders, especially those that live in deep water (on which, I am doing a seminar on this topic).
Plants have another problem that they use to an advantage that the other creatures I mentioned do not have - they have a cell wall - which provides protection from predators and osmotic changes, but limits their ability to move (although not completely). So plants are not exactly able to create muscle-like cells without giving up their cell wall - which is more of a disadvantage according to what we see (but there are some that did - oocytes - fungus-like algae that cause diseases in plants [the main cause of diseases in plants]).
And if we even have to talk about the fact that jellyfish, Echinodermata and Slime moles (which do not have bilaterian symmetry), have the ability to move, and some of them are even more effective in their ability to move than bilaterian creatures?
To conclude - plants do not move because of physiology (cell wall, symmetrical shape) and survival (energy consumption, protection) constraints.
Symmetry has no significant effect on a creature’s ability to move, although there are exceptions or evolutionary constraints that lead to it.
Or alternatively - just luck - we don’t really know why it happened that way - maybe there is a certain advantage, or it’s just luck that it happened that way.