I tall silk plant to the left and a wood bookcase to the right could look really nice

I share the same problem, perfectionism; or is this a gift?! hmm....Anyways, I would suggest placing shelves on only the right side spanning from the fireplace wall to the exterior wall but unfortunately there is a window there. Stating so, a shallow bookcase on the right side of the fireplace would look nice and give it an asymmetrical look or you can install tile / ledge stone around the fireplace, floor-to-ceiling, to create a dramatic and beautiful accent piece in the room!

Put a display piece of furniture to the right , nice period chair , leave the left alone , you cannot get symmetry now so make it look like you have created the extra space to display a piece , book shelf won't work

My world opened up ever since I heard a designer say (and saw what he did) to treat elements that typically are focal points but for whatever reason don't work - an off-placed fireplace (or asymmetrical like yours) and any other design feature - as just another design element. The asymmetrical fireplace would then be a room feature, but not a focal point, so basically you decorate around it.

This really helped me with my old house with almost the exact situation. What you do will depend on how you decorate the rest of the room - for me, I placed an interesting table with a lamp (I needed the lighting) and 2-3 decorative pieces to the smaller space on the left. I had bookcases to the right (some books but not too many + more edited decorative pieces) and had a strong picture and attractive glass that drew the eye to the mantel. The stuff to the left and right were the background, although even the fireplace itself was in the background - the sofa and chairs were centered to the room, not the fireplace or it would have interrupted the flow and been off center. Where you place the rug will help, as will what you use to draw the eye to the other side/rest of the room. HTH.

If the fireplace were symmetrical along the wall, you would never be able to work this room as the furniture to the right of the fireplace would get in the way of the swinging door. In other words, the fireplace is symmetrical as is – it's symmetrical to where the furniture placement will be. Think of the space by the door as a sort of entryway, almost another zone.

Putting a bookshelf or something like that to the right is an effort to draw that entry zone into the seating area, which would be a bit of a mess. To me, space would THEN look asymmetrical.

Forgot to mention in my earlier post that my bookshelf was a built-in and just 1/2 size (around 3' tall). A taller bookshelf in the corner probably would look "off" as other posters have said. You ceilings look to be a couple of feet higher than mine, so not sure. You could pick up a cheap bookshelf from Ikea and see what it looks like for placement before you decide whether to put a decent wood one in.

Hearth and facing should be brighter / louder...distract the eye from lack of symmetry and draw you in to fireplace as a feature.

Do NOT decide quickly. Find your purpose for that corner & then decorate. eg- a fireside chair , a reading nook or a bar cart.