Immortals possess Nascent Souls, which can reincarnate, be reborn, and even reshape the Immortal Body.
Of course, one can cultivate anew after reincarnation or rebirth. However, this is also extremely dangerous because the process of reincarnation requires passing through The Nether World. Unless you have connections in The Nether World, it is not impossible to end up reincarnated as pigs or dogs.
As for rebirth, it's relatively simpler, which involves finding a pregnant woman and then entering her womb to overpower the fetus' spirit and flesh. But after being born, you will be a newborn and might not recover your original cultivation level or memories after re-cultivation.
