The source I have quoted below gives an example of the following stenographic image:-

Is this perception a particular trick that my eye performs or is it processing the visual data in an alternative way?
Stereograms can be viewed as three-dimensional images by providing two
side-by-side views of a three-dimensional scene, rendered from
slightly different viewpoints. Stereoscopes facilitate focusing of the
eyes depths other than the surface of a two-dimensional stereogram
image, allowing the two separate images to overlap in the center of
the field of vision (Kimmel, 2002). Once the correct depth is
achieved, it may seem to the viewer as if the two images are different
due to the horizontal retinal disparities. Then the two images can
fuse into a coherent three- dimensional percept where horizontal
disparities in the images provide important depth cues.

Is the effect of this percept measurable by an EEG?
Autostereograms have been used in recent research to investigate
three-dimensional depth perception. These efforts have improved
understanding of the various visual cues involved in forming
three-dimensional percepts, perceiving motion in depth, attention-
switching in the depth of field, and the neural correlates of forming
and maintaining a unified three-dimensional structure in awareness.
Revonsuo and colleagues (Revonsuo, et al., 1997) used neural
synchronization to investigate the neural correlates of
three-dimensional percept formations. They confirmed that
autostereoscopic perception is primarily mediated by right
parieto-temporal activity. Their research indicated a region of
synchrony spanning portions of the right temporal, parietal and
occipital lobes, with maximum power in the gamma (36-44Hz) frequency
range. Also employing electroencephalogram (EEG) recording, Burgess,
Rehman, and Williams (2003) located particular brain regions,
including the left frontal (F7) and the right and midline occipital
(Oz, O2) clusters, exhibiting functional clustering during
autostereoscopic perception. Overall, neural complexity was highest
with the most functional clustering of responses during observer
awareness of the three-dimensional object. Additional clustering
analyses revealed functional clusters of scalp recordings in the
aforementioned areas that correlated with both phases of
three-dimensional perception (search to three-dimensional and
three-dimensional maintenance).
Sources
Autostereograms and Autostereoscopic Perception
Authors: Leslie Blaha, Christopher Honey, and Eric Nichols
Date: October 31, 2005
Burgess, A.P., Rehman, J., & Williams, J.D. (2003). Changes in neural complexity during the perception of three-dimensional images using random dot stereograms. International Journal of Psychophysiology, 48, 35-42.
Kimmel, R. (2002). Three-dimensional shape reconstruction from autostereograms and stereo. Journal of Visual Communication and Image Representation, 13, 324-333.