Prediction is compression, but I am not sure if it is true the other way around.

It's obvious that an accurate predictor enables encoding only the data that the predictor gets wrong.

But a compressor can encode patterns that defy prediction by looking at the data as a whole. It doesn't have to look at everything in sequence as it arrives.

Applying transformations prior to entropy encoding often isn't just 'rearranging into an easier to compresss format' the transformation can be doing the job of peeking into the future. That makes the encoding a whole lot easier, but it is much harder to call it prediction.