I can't figure out how to flip a Mirror Modifier along the Y axis, after it is mirrored.
Would love some advice on how to achieve this, per the below images:
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Mirror modifier mirrors the object's geometry in it's local space, so if you select X axis and the geometry is towards the +X direction from the origin of the object it will be mirrored to the -X direction, if the geometry is towards the -X direction from the origin, it will get mirrored to the +X direction. You can change the origin point using transform tools if you have Affect Only Origins enabled in the Active Tool and Workspace settings tab in the Properties Editor:
By manipulating location and rotation of the origin(so object's local space) you can control the mirror modifier. You could also use Mirror Object and in that case the modifier will use the object space of that object:
In your specific case, it seems that you do not want to mirror the object so mirror modifier is not suitable. You could duplicate linked (shift+d) and have another object with the same geometry data and just rotate it the way you want, or you could use the new GeometryNodes modifier and rotate a copy of the geometry: