Elegant way of converting between StringComparison and StringComparer?

There isn't something out of the box, but you can create a simple mapping yourself between the comparsion and the comparer:

Dictionary<StringComparison, StringComparer> comparsionToComparer = 
                        new Dictionary<StringComparison, System.StringComparer>
{
    { StringComparison.CurrentCulture, StringComparer.CurrentCulture },
    { StringComparison.CurrentCultureIgnoreCase, StringComparer.CurrentCultureIgnoreCase },
    { StringComparison.InvariantCulture, StringComparer.InvariantCulture },
    { StringComparison.InvariantCultureIgnoreCase, StringComparer.InvariantCultureIgnoreCase },
    { StringComparison.Ordinal, StringComparer.Ordinal },
    { StringComparison.OrdinalIgnoreCase, StringComparer.OrdinalIgnoreCase }
}

And when-ever you need the proper comparer:

var invariantComparer = comparsionToComparer[StringComparsion.InvariantCulture];

Edit:

With C#-6 Dictionary Initializer syntax:

Dictionary<StringComparison, StringComparer> comparsionToComparer = 
                            new Dictionary<StringComparison, System.StringComparer>
{
    [StringComparison.CurrentCulture] = StringComparer.CurrentCulture,
    [StringComparison.CurrentCultureIgnoreCase] = StringComparer.CurrentCultureIgnoreCase,
    [StringComparison.InvariantCulture] = StringComparer.InvariantCulture,
    [StringComparison.InvariantCultureIgnoreCase] = StringComparer.InvariantCultureIgnoreCase,
    [StringComparison.Ordinal] = StringComparer.Ordinal,
    [StringComparison.OrdinalIgnoreCase] = StringComparer.OrdinalIgnoreCase
};

Also, Jons answer refers to the issue of the threads current-culture, which i discarded and should probably be taken into account

Going from StringComparison to StringComparer is simple - just create a Dictionary<StringComparison, StringComparer>:

var map = new Dictionary<StringComparison, StringComparer>
{
    { StringComparison.Ordinal, StringComparer.Ordinal },
    // etc
};

There is a StringComparer for every StringComparison value, so that way works really easily. Mind you, StringComparer.CurrentCulture depends on the current thread culture - so if you populate the dictionary and then modify the thread's culture (or do it from a different thread with a different culture) you may end up with the wrong value. You potentially want a Dictionary<StringComparison, Func<StringComparer>>:

var map = new Dictionary<StringComparison, Func<StringComparer>>
{
    { StringComparison.Ordinal, () => StringComparer.Ordinal },
    // etc
};

Then you can get a comparer at any time by invoking the delegate:

var comparer = map[comparison]();

Going the other way is infeasible, because not every StringComparer has a suitable StringComparison. For example, suppose I (in the UK) create a StringComparer for French (StringComparer.Create(new CultureInfo(..., true)). Which StringComparison does that represent? It's not correct for the current culture, the invariant culture, or ordinal comparisons.