How to convert JSON to XML or XML to JSON?

Yes. Using the JsonConvert class which contains helper methods for this precise purpose:

// To convert an XML node contained in string xml into a JSON string   
XmlDocument doc = new XmlDocument();
doc.LoadXml(xml);
string jsonText = JsonConvert.SerializeXmlNode(doc);

// To convert JSON text contained in string json into an XML node
XmlDocument doc = JsonConvert.DeserializeXmlNode(json);

Documentation here: Converting between JSON and XML with Json.NET

Yes, you can do it (I do) but Be aware of some paradoxes when converting, and handle appropriately. You cannot automatically conform to all interface possibilities, and there is limited built-in support in controlling the conversion- many JSON structures and values cannot automatically be converted both ways. Keep in mind I am using the default settings with Newtonsoft JSON library and MS XML library, so your mileage may vary:

XML -> JSON

1. All data becomes string data (for example you will always get "false" not false or "0" not 0) Obviously JavaScript treats these differently in certain cases.
2. Children elements can become nested-object {} OR nested-array [ {} {} ...] depending if there is only one or more than one XML child-element. You would consume these two differently in JavaScript, etc. Different examples of XML conforming to the same schema can produce actually different JSON structures this way. You can add the attribute json:Array='true' to your element to workaround this in some (but not necessarily all) cases.
3. Your XML must be fairly well-formed, I have noticed it doesn't need to perfectly conform to W3C standard, but 1. you must have a root element and 2. you cannot start element names with numbers are two of the enforced XML standards I have found when using Newtonsoft and MS libraries.
4. In older versions, Blank elements do not convert to JSON. They are ignored. A blank element does not become "element":null

A new update changes how null can be handled (Thanks to Jon Story for pointing it out): https://www.newtonsoft.com/json/help/html/T_Newtonsoft_Json_NullValueHandling.htm

JSON -> XML

1. You need a top level object that will convert to a root XML element or the parser will fail.
2. Your object names cannot start with a number, as they cannot be converted to elements (XML is technically even more strict than this) but I can 'get away' with breaking some of the other element naming rules.

Please feel free to mention any other issues you have noticed, I have developed my own custom routines for preparing and cleaning the strings as I convert back and forth. Your situation may or may not call for prep/cleanup. As StaxMan mentions, your situation may actually require that you convert between objects...this could entail appropriate interfaces and a bunch of case statements/etc to handle the caveats I mention above.

You can do these conversions also with the .NET Framework:

JSON to XML: by using System.Runtime.Serialization.Json

var xml = XDocument.Load(JsonReaderWriterFactory.CreateJsonReader(
    Encoding.ASCII.GetBytes(jsonString), new XmlDictionaryReaderQuotas()));

XML to JSON: by using System.Web.Script.Serialization

var json = new JavaScriptSerializer().Serialize(GetXmlData(XElement.Parse(xmlString)));

private static Dictionary<string, object> GetXmlData(XElement xml)
{
    var attr = xml.Attributes().ToDictionary(d => d.Name.LocalName, d => (object)d.Value);
    if (xml.HasElements) attr.Add("_value", xml.Elements().Select(e => GetXmlData(e)));
    else if (!xml.IsEmpty) attr.Add("_value", xml.Value);

    return new Dictionary<string, object> { { xml.Name.LocalName, attr } };
}

I'm not sure there is point in such conversion (yes, many do it, but mostly to force a square peg through round hole) -- there is structural impedance mismatch, and conversion is lossy. So I would recommend against such format-to-format transformations.

But if you do it, first convert from json to object, then from object to xml (and vice versa for reverse direction). Doing direct transformation leads to ugly output, loss of information, or possibly both.