Things Which Already Have Time-Symmetric Life Cycles
We call them bacteria. The start as a cell, and die as a cell. That seems fairly symmetrical to me!
Alternatively, you could look at viruses, if you consider them to be alive; they start off as a series of proteins which get assembled according to a plan, go out of a cell, travel around, go into a cell, and get disassembled and turn into a plan, which starts the cycle over again. That seems pretty symmetric to me.
As an honorable mention, there is also the immortal jellyfish, whose life cycle, after a certain stage, becomes cyclical, which can be argued to be time-symmetric. From a polyp they become a medusa, and from a medusa (when certain environmental conditions are met) turns back into a polyp. It's not truly symmetric, as they don't go back into a sperm/egg.
Time-Symmetric Life Cycles May Be A Disadvantage
There is hardly any advantage to going back to a sperm/egg after you've spent so much energy becoming the big, multicellular creature that you are. This is likely why so few creatures do this. It is much more efficient to grow up once, and then pass on genes to offspring as much as you can. Time-asymmetric lives are evolutionarily advantageous; a creature does not waste energy and time going back to a little creature, they just go on until something stops them.
There is an argument that going back a younger state may allow a creature to continue living, because maybe a younger state (like a caterpillar) can acquire food better. This is just not truly time-symmetric, though, because it doesn't go down to a single cell once again. So some back-and-forth within a life cycle could be advantageous, as seen in the immortal jellyfish, but it seems rare on earth.
Now, I like to think there is a nice bundle of intelligence within every bacterium, although they are pretty dumb. Often only marginally better than rocks. Some bacteria can evade predators, signal each other, and some are active hunters, but many of them are living on the whims of chance.
As for more advanced intelligence, it would be unlikely that a creature would retain intelligence as it joins back down into a single cell. It would require some exotic scheme to retain that intelligence, like chemical packets serving the same function as memories or neurons, which seems very unlikely!