If matter creates energy and energy creates matter can one exist without the other?
I suppose you're refering to the famous relationship of $E=mc^2$, which expresses the connection between rest mass (that is, mass of an object not moving in our inertial frame of reference) and the energy it possesses. However, how you describe their relationship is not exactly accurate, so let me try to give you a clearer picture.
First of all, there's not only one type of energy but a wide variety of expressions. For example, an object that is moving possesses a kinetic energy, that is, energy related to movement. The most familiar case is an object with mass $m$ that moves with a certain velocity $v$, which has an energy equal to:
At first glance, this may lead you to think only objects with mass can have kinetic energy. However, a photon, which is a particle without mass, can also have a kinetic energy which depends on its frequency $\nu$:
In this case, you can have something with kinetic energy both either with and without mass. You haven't created or destroyed matter in both cases, but the object carries an energy related to its movement. In a similar fashion, you may find energy related to other interactions, such as gravitational potential energy, thermic energy, electromagnetic potential energy, etc.
In the broadest sense, energy is a property of an object that can be transformed and/or transferred to perform some work (i.e. you can convert kinetic energy to push something forward on a collision).
On the other hand, going back to the connection of $E=mc^2$, what this implies is that, given a particle with mass $m$ at rest, totally isolated from its surroundings and without any additional form of energy, if annihilated (say, by meeting its antiparticle) will produce an energy exactly equal to $E=mc^2$. In other words, if an object is annihilated, the mass doesn't simply disappear, but it's transformed into yet another type of energy (in a particle-antiparticle annihilation, this energy is carried by photons, which are massless, with kinetic energy $E=h\nu$). In other words, this means that matter AND energy must be conserved in any interaction (which is a generalization of the law of conservation of energy you may be familiar with).
In the same sense, you can take a massless photon with certain energy near a nucleus, and produce a particle-antiparticle pair. Now, before you didn't have mass, and after the interaction you have mass, so part of the energy of the photons must have been transformed into the mass of the particles by the same relationship of $E=mc^2$. This is the usefulness of this relationship*.
However, again, remember that this is just one type of energy, and there're several expressions of energy you can consider. It's not like they cannot exist without each other.
*Recall again that this equation only applies for particles in rest, there's a more general expression for moving particles but I didn't want to introduce it now as not to confuse you.
Energy can exist without mass: photons have energy but no mass, and the electromagnetic field has an energy density but is generally considered not to have mass. However, mass cannot exist without energy. Mass is energy that is confined.
If a photon were confined in a massless box of perfect mirrors (not physically possible, but let's pretend!), then its energy would be confined and it would have a mass equal to $m$: $m = E/c^2$.