In particular, I want to know if a creature whose upper body displays bilateral symmetry and whose lower body displays tetramerism like a jellyfish is biologically possible.
How about 3 types of symmetry: Bilateral, radial and then tertiary bilateral?
The wily sea cucumber has folded its echinoderm radial symmetry up to recreate bilateral symmetry of its distant ancestors.
Echinoderms are generally radially symmetric, with adults displaying a secondary pentaradial symmetry. The symmetry is secondary, because echinoderm larvae are bilaterally symmetric. One group, the sea cucumbers, developed a tertiary bilateral symmetry.
During metamorphosis, as in other echinoderms, the larva of an holothuroid changes from bilateral to pentaradiate symmetry. This is seen mainly in the radial structure of the water vascular system, including tube feet and associated tentacles. Superimposed on this adult radial symmetry, however, is a bilateral symmetry, evident externally in the division of the body into a ventral trivium and dorsal bivium, and internally by the bilateral nature of the complex hemal system and associated gut tube. Traditional thinking is that phylogenetically, holothurians derived from ancestors with a primary radial symmetry and later adopted a crawling habit of life with accompanying secondary bilaterality. The later imposition of radial symmetry is therefore generally considered a tertiary event in their evolution.
It should be very easy for an organism with even-numbered radial symmetry over part of its body to display bilateral or x-lateral symmetry (in which x is a factor of the superior even number) elsewhere. As long as both structures are divisible by the lower-order symmetry, the organism can be classified by that degree of symmetry.
I don't see any reason why not.
While not a true example, an octopus or other cephalopod has bilateral symmetry for their mantel, and arms and tentacles that radiate out around their mouth. They are in fact only bilaterally symmetrical.
There are also starfish that have bilateral symmetry as larva and radial symmetry as adults.
Yes, but it most likely wouldn't be perfect/near-perfect.
As an example, take humans. Our hands have five-fold symmetry: if someone took a picture of a finger without any of the rest of the hand in the frame, it might be tough to tell which one it was a picture of. That's translational symmetry. However, there's a break in the symmetry: the thumb and pinky each only have one neighbor, and thus their roles are different, and they look different. It would be inefficient for parts which are in different positions to be identical, and inefficient in a way that's easily optimized by evolution.
The organs and limbs of your organism would be similar; in this case the break in symmetry is that the quadrants have different positions relative to the bilaterally symmetric part of the organism. Therefore, the back and front quadrants (or the back quadrant, front quadrant, and two side quadrants) would be different from each other, on a level somewhere around how different your thumb or pinky is from your pointer finger.
A sphere has all types of symmetry simultaneously. And of course, an organism can be spherical.
You mean one part having only one type of symmetry and the other part having another one type of symmetry. Yes, only spherical symmetry cannot participate in that.
The upper part having two symmetrical hands and the lower - three symmetrical legs? It is possible, too. Imagine a Minopode - bull's head, human upper body and octopus' tentacles instead legs.