Update:

Here's the full script I'm using for displaying the device camera feed on a UI image with correct orientation and ratio, front and back cameras, and mirroring the front camera to make it look more natural, along with a screenshot of my hierarchy and RawImage game object. It's based on this answer by @Fattie, this other answer by Max Bot, reading the Unity documentation on WebCamTexture and WebCamDevice, and trial and error on a Nexus 5 running Android 6 and an iPhone 5S running iOS 9.

Hierarchy:

DeviceCameraController.cs:

 using UnityEngine;
 using UnityEngine.UI;
 using System.Linq;
 using System.Collections;
 
 public class DeviceCameraController : MonoBehaviour
 {
     public RawImage image;
     public RectTransform imageParent;
     public AspectRatioFitter imageFitter;
 
     // Device cameras
     WebCamDevice frontCameraDevice;
     WebCamDevice backCameraDevice;
     WebCamDevice activeCameraDevice;
 
     WebCamTexture frontCameraTexture;
     WebCamTexture backCameraTexture;
     WebCamTexture activeCameraTexture;
 
     // Image rotation
     Vector3 rotationVector = new Vector3(0f, 0f, 0f);
 
     // Image uvRect
     Rect defaultRect = new Rect(0f, 0f, 1f, 1f);
     Rect fixedRect = new Rect(0f, 1f, 1f, -1f);
 
     // Image Parent's scale
     Vector3 defaultScale = new Vector3(1f, 1f, 1f);
     Vector3 fixedScale = new Vector3(-1f, 1f, 1f);
 
 
     void Start()
     {
         // Check for device cameras
         if (WebCamTexture.devices.Length == 0)
         {
             Debug.Log("No devices cameras found");
             return;
         }
 
         // Get the device's cameras and create WebCamTextures with them
         frontCameraDevice = WebCamTexture.devices.Last();
         backCameraDevice = WebCamTexture.devices.First();
 
         frontCameraTexture = new WebCamTexture(frontCameraDevice.name);
         backCameraTexture = new WebCamTexture(backCameraDevice.name);
 
         // Set camera filter modes for a smoother looking image
         frontCameraTexture.filterMode = FilterMode.Trilinear;
         backCameraTexture.filterMode = FilterMode.Trilinear;
 
         // Set the camera to use by default
         SetActiveCamera(frontCameraTexture);
     }
 
     // Set the device camera to use and start it
     public void SetActiveCamera(WebCamTexture cameraToUse)
     {
         if (activeCameraTexture != null)
         {
             activeCameraTexture.Stop();
         }
             
         activeCameraTexture = cameraToUse;
         activeCameraDevice = WebCamTexture.devices.FirstOrDefault(device => 
             device.name == cameraToUse.deviceName);
 
         image.texture = activeCameraTexture;
         image.material.mainTexture = activeCameraTexture;
 
         activeCameraTexture.Play();
     }
 
     // Switch between the device's front and back camera
     public void SwitchCamera()
     {
         SetActiveCamera(activeCameraTexture.Equals(frontCameraTexture) ? 
             backCameraTexture : frontCameraTexture);
     }
         
     // Make adjustments to image every frame to be safe, since Unity isn't 
     // guaranteed to report correct data as soon as device camera is started
     void Update()
     {
         // Skip making adjustment for incorrect camera data
         if (activeCameraTexture.width < 100)
         {
             Debug.Log("Still waiting another frame for correct info...");
             return;
         }
 
         // Rotate image to show correct orientation 
         rotationVector.z = -activeCameraTexture.videoRotationAngle;
         image.rectTransform.localEulerAngles = rotationVector;
 
         // Set AspectRatioFitter's ratio
         float videoRatio = 
             (float)activeCameraTexture.width / (float)activeCameraTexture.height;
         imageFitter.aspectRatio = videoRatio;
 
         // Unflip if vertically flipped
         image.uvRect = 
             activeCameraTexture.videoVerticallyMirrored ? fixedRect : defaultRect;
 
         // Mirror front-facing camera's image horizontally to look more natural
         imageParent.localScale = 
             activeCameraDevice.isFrontFacing ? fixedScale : defaultScale;
     }
 }

Comments on Fattie's Answer:

So I'm pretty sure at this point that "verticallyMirrored" means that a rotation about the horizontal axis is needed. So I've modified the code in Fattie's answer to:

    int ccwNeeded = -wct.videoRotationAngle;
    
    rawImageRT.localEulerAngles = new Vector3(0f,0f,ccwNeeded);
    
    float videoRatio = (float)wct.width/(float)wct.height;
    
    rawImageARF.aspectRatio = videoRatio;
    
    if ( wct.videoVerticallyMirrored )
      rawImage.uvRect = new Rect(0,1,1,-1);  // flip on HORIZONTAL axis
    else
      rawImage.uvRect = new Rect(0,0,1,1); // no flip

i.e. I removed "if ( wct.videoVerticallyMirrored ) ccwNeeded += 180;" and changed the flipped uvRect values. I'll post a new answer here or somewhere on this site with my full script once I iron it out which includes using both front and back cameras and mirroring the front camera's texture to make it more natural.

Have you tried NatCam? It works wonders and has a lot more features.

Unity 5.6.1 Release Notes [Extract]:

• Android: Fixed WebCamTexture crash with denied permissions. (877837)

• Android: Webcam - Fixed the wrong orientation returned on first frames. (875247)

Hope this solves a small issue faced here.