In a typical refraction example, say air for y>0 and water (denser) for y<0, a light ray from air bends at the air-water interface and continues in generally the same direction: that is, the direction vectors of the incident and refracted rays are in the same quadrant. (If the incident ray comes in from the upper left, (-x,+y), and goes rightward and downward towards the origin at (0,0), the refracted ray goes out (+x,-y), also rightward and downward. The angles are measured between the -y axis (the normal to the interface) and the ray direction vectors.)
Snell's Law says that Ni sin(theta_i) = Nr sin(theta_r), where i is for the incident ray (in air), r is for the refracted ray (in water), and the N's are the indices of refraction.
Since sin(-theta_r) = -sin(theta_r), I would think that a negative refraction angle should represent a ray going out with one component reversed, e.g. downward, but leftward. (The alternative would be upward and rightward, which would be a reflection back into the air.) The refracted ray would point in a different quadrant. For example, if you were standing on the moving end of a diving board and looked down into the water, you normally see the bottom of the pool in front of you; but, with a negative refraction angle, you would look forward and down and see the pool floor under the diving board and distorted.