Unsure of how to prevent 270 degrees from being confused with 90 degrees when converting from rotational matrices to quaternion

Question

I have a struct for a 3x3 rotational matrix of integers that I need to store the data for rotation of objects in a 3D environment to be easily serialized into XML and interpreted by another program

TL;DR, Unsure of how to implement a fix for a quaternion of 270 degrees resulting in 90 instead

public struct IntegerMatrix3x3
    {
        public int m00;
        public int m10;
        public int m20;

        public int m01;
        public int m11;
        public int m21;

        public int m02;
        public int m12;
        public int m22;

        /*
        m00 m01 m02 
        m10 m11 m12 
        m20 m21 m22
        */
    }

The issue I have come across is a with a function I have made to convert the matrix to a vector of Euler angles I can use to set the game objects transform in the unity environment

public Vector3 ToEulers()
        {
            // NTS: http://www.euclideanspace.com/maths/geometry/rotations/conversions/matrixToQuaternion/index.htm
            Quaternion q = new Quaternion();
            q.w = Mathf.Sqrt(Mathf.Max(0, 1 + m00 + m11 + m22)) / 2;
            q.x = Mathf.Sqrt(Mathf.Max(0, 1 + m00 - m11 - m22)) / 2;
            q.y = Mathf.Sqrt(Mathf.Max(0, 1 - m00 + m11 - m22)) / 2;
            q.z = Mathf.Sqrt(Mathf.Max(0, 1 - m00 - m11 + m22)) / 2;
            q.x *= Mathf.Sign(q.x * (m21 - m12));
            q.y *= Mathf.Sign(q.y * (m02 - m20));
            q.z *= Mathf.Sign(q.z * (m10 - m01));
            // Find euler angles from quaternion derived from matrix
            Debug.Log(q.eulerAngles);
            return q.eulerAngles;  
        }

When I encode the angles (0, 270, 0) into the matrix, then convert the matrix back to a angles using said function, it outputs (0, 90, 0). I did a bit of reading and discovered that that is a downfall of quaternions/matrices, but I am not sure how to implement a solution/workaround for it.

Thanks, Nifley