Is this animated gif an accurate representation of EM waves?

These look like TE waves in a rectangular waveguide. The electric field, magnetic field and the direction of propagation form a mutually orthogonal set (TEM wave) in free space propagation , not necessarily in guided waves. In fact, one can show from Maxwell's equations that a single conductor waveguide cannot support a single one of those TEM waves you are talking about. These structures support TE and TM waves, which have magnetic and electric fields along the propagation direction, respectively.

The fields in that gif are the TE10 mode of a rectangular waveguide which look like this:


(Image from Field and Wave Electromagnetics by D.k.Cheng)

I believe that is a visualization of a wave confined to a waveguide, not travelling in free space. This mode is called a TE mode of the waveguide because the E field is perpendicular to the axis of the waveguide but the H field is not.

Because the field goes to zero beyond certain boundaries in the transverse direction, we can guess this is meant to be a wave travelling in a waveguide with perfectly conducting boundaries (as opposed to, for example, a dielectric waveguide).

The reason the H field is not perpendicular to the overall direction of power flow is that there is essentially a superposition of two waves here, one zig-zagging left and right across the waveguide and the other zig-zagging right and left.