Are there sounds in English languages and accents where the tongue does not move symmetrically in the mouth, i.e. the right side of the tongue is not moving like the left side?
It's extremely rare for any language to have a sound where the target pronunciation of that sound involves an asymmetric tongue position. In fact, I'm unaware of an example (although I wouldn't like to say that absolutely none exists).
But in practice, it's very common (in languages in general, not just English) for tongue contact to be asymmetric. For example, in the pronunciation of laterals (such as English "l" sounds), which are canonically defined as having the air escape "at the sides", in practice there may be tongue contact at one side with air escaping at only one side. Similarly, in alveolar and palatal stops generally, there may be more tongue contact at one side than the other.
If you're interested more in this subject, then take a look at any study where palatograms have been taken: these give a "map" of tongue contact as sounds are produced. (Specifically, the modern type are termed electropalatograms, produced from an array of contact-detecting electrodes embedded in a false palate. An earlier primitive technique involved a false palate covered in chalk.)
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The only one I can think of is the lateral click, which occurs in some South African languages.
It is known to English speakers as a sound used to gee up horses, but is not part of the English language.
Considering the English Phonetic system, and if you mean watching the tongue from the front, the answer is no.
The lateral (L) is called like this because the air flows through the sides. It can happen that you, while speaking, put the tongue in some weird positions but that is your realisation, due to the situation, position, etc.
It's possible that in other languages there are realisations where the tongue is not symmetrical, but I'm not aware of any in English.
Try looking at the IPA for the English phonemes and try to "say" them. Your tongue will change position but, basically, it will always remain symmetrical (not mathematically symmetrical, of course, but you know what I mean).