Will I even encounter any enharmonic chords? For example would an inverted variation of one chord be equivalent to the root variation of another chord?
Yes they exist. I don't know all of them of the top of my head, but I'll give you the three that come to mind easily.
The first is a fully diminished chord. Because there are only 12 named notes and a fully diminished chord is made of 4 notes that are a minor 3rd apart from each other (3 semitones) there are only 3 different chords but each can be named 4 different ways. For example, a C fully diminished chord would consist of the notes
C-Eb-Gb-Bbb, a Eb fully diminished chord would consist of the notes
Eb-Gb-Bbb-Dbb(C), a Gb fully diminished chord would consist of the notes
Gb-Bbb-Dbb-Fbb(Eb), and an A (Bbb) fully diminished chord would consist of the notes
The second is an augmented chord and like the fully diminished chord because there are only 12 named notes and an augmented chord chord is made of 3 notes that are a major 3rd apart from each other (4 semitones) there are only 4 different chords but each can be named 3 different ways. For example a C augmented chord would consist of the notes
C-E-G#, an E augmented chord would consist of the notes
E-G#-B#(C), and a G# augmented chord would consist of the notes
The third example are 7th and 6th chords. Some minor 7th chords contain the same notes as 6th chords. For example a C6 would contain the notes
C-E-G-A and an Am7 would contain the notes
In all these examples you name the chord based on the context of the key for example if you are in the key of Db minor and you see the notes Eb, C, Gb, and Bbb you would call that a C fully diminished chord because C is the leading tone of Db minor so it makes the most sense in the key.
Another example is there are polychords and extensions that will contain the same notes. A very simple example is Eb/C (produced Eb over C) can be represented as an C7#9. While the notes of each are the same, a polychord refers to a specific idea where there are two distinct perceived chords rather then just one.
I will avoid the debate in the comments about whether "enharmonic" is the correct word.
But yes there are loads of examples of chords with the same notes in them.
C D G = Csus2
G C D = Gsus4
A C E G = Am7
C E G A = C6
The full diminished and augmented chords where any one of the notes can be considered a root without even changing the type of harmony:
B D F Ab/G# (full diminished)
C E ab/G# (augmented)
Example in use and conclusion
The best name depends on context, that is what chord it precedes and follows, more than it does on the inversion used.
The following (rather overused) chord progression I heard recently in a local band's song springs to mind:
C Am F G
At the end of the song it went uptempo and the chord progression changed to the following:
C6 Am7 F G
But the C6 and Am7 were exactly the SAME CHORD, completely blurred together.
EDIT: John Lennon's Happy Christmas (War is Over)
Just occured to me this is a good example of why you need different names for the same chord. If we look at the chord progression over the first half of the verse, it's easy even for a non musician to appreciate the symmetry and realise why this song is so popular. The fast change through chord type (3rd, sus2, sus4 and back to 3rd) combined with the slow change through chord root (A B E A) make this song both energising and soothing at the same time.
A Asus2 Asus4 A
Bm Bsus2 Bsus4 Bm
E Esus2 Esus4 E
A Asus2 Asus4 A
Now, Esus4 and Asus2 contain the same notes. If I wanted to standardise the name of this chord I could pick Asus2. Applying this rigorously I get
A Asus2 Dsus2 A
Bm Bsus2 Esus2 Bm
E Esus2 Asus2 E
A Asus2 Dsus2 A
Which is totally confusing and completely misses the point of the chord progression.
There are chords that function in different ways in different contexts. Take for example the chord C-E-G-A. This is the tonic chord of the key of C major with an additional note. It is called an "added-sixth" chord. It has a very pleasing sound and is often used as an ending chord in jazz and in the music of Olivier Messiaen.
Now play C-E-G-A on your keyboard, but follow it with C-D-F#-A and then B-D-G. As if by magic the chord has become the first inversion of the ii7 of G major; the "dominant seventh of the dominant seventh". Same notes, but what was a restful ending chord is now a chord that has to move in order to resolve.
I realize that this is in some sense the dual to what you asked, but here's one of the most famous chords in music history, the Tristan Chord. 150 years after it was first played, still nobody can agree on how to interpret (and thus name) that chord.
I don't think there are nor have I ever encountered any. It is mainly because when the notes in a chord are being played, no matter in what order, they all give the same feel as if they are being played together (due to the delay of the previous notes in your head). There are "enharmonic scales" though, for example harmonic minor scale (1 2 b3 4 5 b6 7) and phrygian dominant scale (5 b6 7 1 2 b3 4) both consist of the same notes.