c# Can a "task method" also be an "async" method?

You may want to read my async/await intro post.

Return values from async methods are wrapped in a Task<TResult>. Likewise, await unwraps those return values:

public static async Task<String> LongTaskAAsync() {
  await Task.Delay(2000);
  return await Task.Run(() => {
    return("AAA");
  });
}

The reasoning behind this is described in my Async "Why Do the Keywords Work That Way" Unofficial FAQ.

P.S. You can also use Task.FromResult for simple tests like this.

Edit: If you want to create and return the Task object itself, then the method should not be async. One somewhat common pattern is to have a public non-async method that calls the async portion only if necessary.

For example, some kind of asynchronous cache - if the object is in the cache, then return it immediately; otherwise, asynchronously create it, add it to the cache, and return it (this is example code - not thread-safe):

public static Task<MyClass> GetAsync(int key)
{
  if (cache.Contains(key))
    return Task.FromResult(cache[key]);
  return CreateAndAddAsync(key);
}

private static async Task<MyClass> CreateAndAddAsync(int key)
{
  var result = await CreateAsync(key);
  cache.Add(key, result);
  return result;
}

Can a “task method” also be an “async” method?

Yes it can be, by simply changing the method signature to public async static Task<Task<String>> LongTaskAAsync() since that is, what it will return.

If you use the async keyword, the runtime will wrap the type you return into a task, to enable asynchronousness. Say if you return a string, the runtime will wrap that into a Task<string>. int will go Task<int> and Task<string> will go Task<Task<string>>. See this console app to clearify:

public class Program
{
    public static void Main(string[] args)
    {
        // start the main procedure asynchron
        Task.Run(() => DoIt()).Wait();
    }

    // for async support since the static main method can't be async
    public static async void DoIt()
    {
        Program p = new Program();

        // use the methods
        string s = await p.GetString();
        int i = await p.GetInt();
        Task<string> tsk = await p.GetTaskOfString();

        // just to prove the task works:

        // C# 5
        string resultFromReturnedTask = await tsk;

        // C# 4
        string resultFromReturnedTask2 = tsk.Result;
    }

    public async Task<string> GetString()
    {
        return "string";
    }

    public async Task<int> GetInt()
    {
        return 6;
    }

    public async Task<Task<string>> GetTaskOfString()
    {
        return Task.Run(() => "string");
    }
}