Inkscape does not have any automatic method which fixes your image to perfectly symmetric. Apply Path > Simplify to reduce small twists.Apply Path > Break apart and delete the unwanted dots here and there.Change stroke color to orange and select narrower stroke width for better visibility.The next image shows the centerline tracing result after some manual tweaks: It doesn't know the magic perfect curve that you obviously can imagine. Unfortunately it will not be as regular as you hope, it's only as regular as you have drawn + how the smoothing adjustment decides. Finally remove the construction lines and the raster image. If you need this object converted to a path, do Path > Object to Path, and then select and join the nodes of each segment using the Join Selected Nodes button.Apply two Mirror Live Path Effects, leave the first at the default, but for the second choose the "Y coorinate mirror" option, and click and drag the mirror line down until it intersects with the origin.Zoom in to get it as close as possible to the original. With the Bézier/Pen tool, draw one curve.Drag a horizontal and vertical guide to these lines using Snapping. Line everything up so that the two centre nodes intersect to create an origin at the exact centre of the shape. With the pen tool, draw two lines to bisect the shape horizontally and vertically, add a node in the middle of each line using the add nodes button in the Pen Tool options.Here's a semi-automatic method to generate a symmetrical shape, which doesn't involve auto tracing, but still requires manually re-drawing one of the curves. I wouldn't bother with auto tracing, as it's sure to introduce irregularities.
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