gdal - Local coordinate system to EPSG via affine transformation

I have a raster file with a local coordinate system. It is based on EPSG:31466 but with a specific rotation and offset.

gdalinfo output is this:

I need to transform this gtiff back to 31466, but how?

With FME I can use the Affiner transformer to do this.

Affiner parameter:

gdalinfo of output geotiff from FME:

But I need to know the workflow with gdal.

I'm using gdal 2.3.0.