Cross Product Inverse of what I want for wall run

I am programming my own FPS wall run in a custom character motor using the Character Controller. It is almost where I want, except for one thing. My interpretation of geometry and math seems to be off. I am trying to get the cross product from the MoveDirection and the normal of the wall to run against which, as far as I know, should get a vector parallel with the wall which I can then use as the character's move direction.

And it does! The thing is, it's in the opposite direction of where I want to go. Debug rays are showing it, and it's just plain obvious in the movement. Here's an image to show what I'm going for, and what is happening (the Magenta line is what's occurring):

I want to get input from outside sources though to make sure this is occurring as expected, or if I'm missing something vital. If it's as expected, then I can do a little more math to get the inverse, but if I'm doing something wrong and don't see it, I should correct it.

What I'm doing is casting a ray out to the right of my character. If it hits something, it then gets the cross-product of the character's MoveDirection and the plane's normal, and sets that as the new direction before the update moves on with the rest of the housekeeping (the full structure I'll tweak once I know I'm doing the right thing.) Here is the function that creates the wall run vector:

     bool DoWallRunCheck(){

         Ray ray = new Ray(transform.position, transform.TransformDirection(Vector3.right));
         Debug.DrawRay(transform.position, transform.TransformDirection(Vector3.right), Color.red, 2f, false);

         RaycastHit hit;

         if (Physics.Raycast(ray.origin, ray.direction, out hit, 1f)){

             Debug.DrawRay(hit.point, hit.normal, Color.green, 2f, false);

             Vector3 crossProduct = Vector3.Cross(moveDirection, hit.normal);
             Debug.DrawRay(transform.position, crossProduct * BaseSpeed, Color.magenta, 2f, false);

             moveDirection = crossProduct;
             moveDirection = transform.TransformDirection(moveDirection);
             moveDirection.Normalize();
             moveDirection *= BaseSpeed + (RunSpeedIncrease * (moveDownTime / RampUpTime));

             return true;
         }
         else {
             return false;
         }
     }

Just to note, I have also tried it with:

 Vector3 crossProduct = Vector3.Cross(moveDirection, hit.normal);
 crossProduct = Vector3.Cross(crossProduct, hit.normal);

That was suggested to me by one source who's better with vector math than I am, but still produces the same result.