Diode in opposite direction?

At the point just before the first amp, the radio waves have been filtered to a particular frequency by L1, C1 & C2. That first amplifier is not an op-amp, it is an RF gain amplifier. It amplifies the incoming signal by a number of dB. The signal before and after will be an AC signal, equally biased around ground.

The diode or detector gets rid of one half of the signal (either the positive half or the negative half, depending on which way the diode is). Description of AM Detector.

The next stage filters off the carrier signal with a low-pass filter.

As the next amp, an op-amp has a bipolar power supply so it can handle either the positive or negative signal. It amplifies the sound waveform for the speaker.

In this particular circuit application, the demodulation of amplitude modulated RF, no, the polarity of the diode does not matter. If you reverse the diode, you simply track the positive part of the envelope rather than the negative part. Either will give you the demodulated signal.