Is it possible to lose half the samples, and yet see the full spectrogram as the original signal?
To rephrase your question: Is there a process that halves the number of samples of a full-bandwidth signal without significantly affecting the appearance of the spectrogram, except for scaling of time or frequency axes?

Resampling does not do that. The only such process I know of is time stretching / pitch scaling followed by decimation. Actually, one definition of time stretching / pitch scaling would be to "modify the signal so that the spectrogram is stretched".

Time stretching and pitch scaling are equivalent from the point of view that the number of samples needed to represent the processed signal changes. Time stretching to half the original length requires only half the number of samples, and pitch scaling to half the frequency allows to discard every second sample without causing aliasing, assuming that the empty upper part of the frequency spectrum is not filled by something made-up by the algorithm used.

Time stretching / pitch scaling does not have a unique definition or a unique mathematical formula. Well-done time stretching should match one's expectation of, in your case, "speeding up" the process that generated the audio. Speeding up natural processes is not uniquely defined. For example, should the pianist be made to move their fingers faster, resulting in changes in note attacks in the least, or should we be more agnostic of the actual process taking place and just isolate the notes in the audio and move them around into a more compact pattern? And should the note decay times be made shorter, too, because we want to make everything shorter? Depends on one's expectations or needs.

There are various algorithms though. Here are before and after spectrograms of the phrase "hello" (freesound), processed by rubberband -t0.5 speech16kHz.wav out.wav with Rubber Band v1.8.2 command line and analyzed in Adobe Audition 3.0 with 512 bands and Blackman–Harris window:



Figure 1. Top: original speech spectrogram, bottom: spectrogram of the speech time-stretched to half the original length and half the number of samples. The spectrograms are zoomed out to full view for easier comparison.

The spectrogram of the time-stretched signal looks essentially the same, but has reduced detail and resolution.