Quaternion[] rot = new Quaternion[] {
    new Quaternion(0,0,0,1),
    new Quaternion(0,0,0.707f,-0.707f),
    new Quaternion(0,0,0.707f,0.707f),

    new Quaternion(0,0,1,0),
    new Quaternion(0,0.707f,-0.707f,0),
    new Quaternion(0,0.707f,0,-0.707f),
    new Quaternion(0,0.707f,0,0.707f),
    new Quaternion(0,0.707f,0.707f,0),

    new Quaternion(0,1,0,0),
    new Quaternion(0.5f,-0.5f,-0.5f,-0.5f),
    new Quaternion(0.5f,-0.5f,-0.5f,0.5f),
    new Quaternion(0.5f,-0.5f,0.5f,-0.5f),
    new Quaternion(0.5f,-0.5f,0.5f,0.5f),

    new Quaternion(0.5f,0.5f,-0.5f,-0.5f),
    new Quaternion(0.5f,0.5f,-0.5f,0.5f),
    new Quaternion(0.5f,0.5f,0.5f,-0.5f),
    new Quaternion(0.5f,0.5f,0.5f,0.5f),

    new Quaternion(0.707f,-0.707f,0,0),
    new Quaternion(0.707f,0,-0.707f,0),
    new Quaternion(0.707f,0,0,-0.707f),
    new Quaternion(0.707f,0,0,0.707f),

    new Quaternion(0.707f,0,0.707f,0),
    new Quaternion(0.707f,0.707f,0,0),
    new Quaternion(1,0,0,0)
  };

int NearestRot(Quaternion cubeRot){
    int nearestIndex = 0;

    for (int i = 0; i < 24; ++i) {
      float maxDot = -10000.0f; //NEG INF
      float dot = Mathf.Abs(Quaternion.Dot(
        cubeRot, rot[i]));
      if (dot > maxDot) {
        maxDot = dot;
        nearestIndex = i;
      }
    }
    return nearestIndex;
  }

void FixedUpdate() {
  cubeRigid.rotation = Quaternion.Slerp(cubeRigid.rotation, rot[NearestRot(cubeRigid.rotation)], 10f*Time.fixedDeltaTime);
}

using UnityEngine;
using System.Collections;
using UnityEngine.UI;

public class quaternionTest : MonoBehaviour {

    // UI иконки включенного направления
    public Image xImg;
    public Image yImg;
    public Image zImg;
    // собственно тело
    public Transform cube;
    // приват перем для переключения напр
    int currentDim = 0;

    void Start () {
        SetActiveDim(1); // сразу включаем направление Х
    }
    
    void FixedUpdate () {
        if (Input.GetKeyDown(KeyCode.Alpha1))
            SetActiveDim(1);
        if (Input.GetKeyDown(KeyCode.Alpha2))
            SetActiveDim(2);
        if (Input.GetKeyDown(KeyCode.Alpha3))
            SetActiveDim(3);

        float msDelta = Input.GetAxis("Mouse ScrollWheel");

        if (msDelta != 0) {
            Vector3 axis = Vector3.zero;

            switch (currentDim) {
                case 1: axis = Vector3.left; break;
                case 2: axis = Vector3.up;   break;
                case 3: axis = Vector3.forward; break;
            }

            cube.Rotate(axis, 10f * msDelta);
        }
    }

    void SetActiveDim(int dim) {
        xImg.color = yImg.color = zImg.color = Color.gray;
        // у меня иконка просто рамка из дефолт спрайта, а в ней буква :)
        Text xTxt = xImg.GetComponentInChildren<Text>();
        Text yTxt = yImg.GetComponentInChildren<Text>();
        Text zTxt = zImg.GetComponentInChildren<Text>();
        xTxt.color = yTxt.color = zTxt.color = Color.gray;

        currentDim = dim;

        switch (dim) {
            case 1:
                xImg.color = xTxt.color = Color.red; // UI элемент
                // тут "выравниваем" оси, перед поворотом
                cube.rotation = Quaternion.Euler(cube.rotation.eulerAngles.x, 0f, 0f);
                break;
            case 2:
                yImg.color = yTxt.color = Color.green;
                cube.rotation = Quaternion.Euler(0f, cube.rotation.eulerAngles.y, 0f);
                break;
            case 3:
                zImg.color = zTxt.color = Color.blue;
                cube.rotation = Quaternion.Euler(0f, 0f, cube.rotation.eulerAngles.z);
                break;
        }
    }
}