Player moving/rotating along a single axis

There's an error in these lines:

     localRotation.y += Input.acceleration.x * speed;
     localRotation.x += Input.acceleration.y * speed;

localRotation is a quaternion, and their x,y,z,w components have nothing to do with the familiar angles we see in the Rotation field in the Inspector - this field is actually transform.eulerAngles. It's a very common mistake, which most people do at first (me included).
In order to keep your logic, you should declare localRotation as a Vector3, and store transform.eulerAngles in it at Start - like below:

using UnityEngine; using System.Collections; public class Playermovement : MonoBehaviour { public float Speed = 16.0f; //left/right movement speed private Vector3 localRotation; // declare localRotation as a Vector3 public float speed = 16.0f; // left/right movement speed public float rollSpeed = 5.0f; public float rollAmount = 10.0f;

 void Update() {
     rigidbody.velocity = new Vector3(0, rigidbody.velocity.y, 0);
 
     transform.Translate(Input.GetAxis("Horizontal") * Time.deltaTime * Speed, 0, 0);

     Vector3 dir = Vector3.zero;  //Android controls
     dir.x = -Input.acceleration.y;
     dir.z = Input.acceleration.x;
     if (dir.sqrMagnitude > 1)
         dir.Normalize();
     
     dir *= Time.deltaTime;
     transform.Translate(dir * Speed);

     // first update the current rotation angles with input from acceleration axis
     localRotation.y += Input.acceleration.x * speed;
     localRotation.x += Input.acceleration.y * speed;
     rollAmount *= 0.9f;

     if (Input.GetKey("left")) {
         rollAmount += rollSpeed ;
     } else if (Input.GetKey("right")) {
         rollAmount -= rollSpeed ;
     }

     transform.Rotate(0, rollAmount*Time.deltaTime, 0);
     rigidbody.freezeRotation = true;
 }  

 // Use this for initialization
 void Start (){
    // copy the rotation of the object itself into a buffer
    localRotation = transform.eulerAngles;
 }

} But you're modifying the Z position in the "Android controls" section, as @You (not you!) said - maybe you (not @You - God, this is becoming very confusing!) should use dir.y instead of dir.z in that section.