String Theory's Graviton and Spin-2 Particle Controversy

According to the current physics dogma, the graviton is a spin-2 particle. One argument for this involves the fact that Einstein's field equations are rank-2 tensors. Rank-2 tensor corresponds with spin-2 particle? If that's the case, then a 1/2-spin electron should correspond with a 1/2-rank tensor--but there is no such tensor! Just when you've given up hope that the graviton is a spin-2 particle, another argument is thrown against the wall to see if it will stick. In the following video, Leonard Susskind presents that argument:

There is a rule that says that left-moving energy must equal right-moving energy along a closed string. Each point along the string (represented by sigma) could serve as a starting point and the end point (equal zero and 2pi); i.e., the closed string and its points are invariant. Because right-moving and left-moving energy adds to zero, and because the invariant string and its points are equivalent, the closed string must have right-moving energy and left moving energy, and those two energies must be equal. Clear as mud, right? But that's essentially the argument in favor of spin-2 particles. Here is how the graviton is represented:

The a's and b's in parenthesis are the creation operators for the x and y axis, respectively. Complex numbers are used to show angular momentum. Notice there are two angular-momentum creation terms; one for left motion and one for right motion. There are two possible states: the two angular momenta going clockwise or counter-clockwise. Now compare this setup to the photon's:

The photon has only one momentum or one spin. There is no right-left motion business. And why should there be? When angular momentum completes a cycle, the total displacement is zero. If there is a vector moving left, there is an equal, opposite vector moving right. No need for an extra spin. An extra spin is redundant. This is why the left-right-motion argument fails to justify a spin-2 particle. Nature only requires one spin to complete a cycle that adds to zero. Here is the math:

Take the integral of the entire cycle of the spin (360 degrees) and you get the equivalent of the right-left-motion business--which is zero.

Bearing all this in mind, does the graviton (assuming it exists) need to be a spin-2 particle?