Background
I respectfully disagree with @lassoan’s comments regarding NIfTI being fragile. From my perspective, the NIfTI format is very simple (just 348 bytes) and explicit, but it is tuned to a specific domain. Relative to other formats like DICOM, NRRD and NIfTI are very similar. One main difference between the two is that to support NRRD you need a very good reader to parse the various spatial frames of reference. On the other hand, NIfTI uses only a single spatial coordinate system so reading is simple, but the writing of NIfTI is challenging as you must use the NIfTI frame of reference.
The biggest issue many have with NIfTI is that it can include two different spatial transforms - the QForm and the SForm. The qform_code and sform_code allow you to understand what they refer to. While usage can vary, the original rationale is that the Q-Form code could map the image alignment to the scanner’s acquisition space, while the S-Form can map the image to a standard space where the participants brain is aligned to a typical brain size, rotation and origin (with a full affine 12 degrees of freedom: 3 axes rotation, 3 axes translation, 3 axes zoom and 3 axes shear). It is particularly unfortunate that the NIfTI creation specified that the Q Form uses a quaternion (which only has 9 degrees of freedom) while the S Form uses a matrix (with 12 DoF). This does complicate support for the format. Also, since the SForm supports shears, where many image viewers require are unable to support rhomboidal images.
The Insight ToolKit (ITK) does not support rhomboidal images. While this may have changed, early versions of ITK always gave precedence to the QForm (presumably because the constrained QForm can not represent rhomboidal images), while most other tools chose between the SForm and QForm by selecting the higher qform_code versus sform_code (and giving precedence to the SForm in the case of ties). This meant that the starting estimates for ITK based tools like ANTS were often very different from other NIfTI tools like SPM and FSL.
For these reasons, most tools today create NIfTI images where the SForm and QForm are identical. Rhomboidal images are virtually never seen in MRI (though they exist with CT scans acquired with gantry tilt), so most tools limit NIfTI images to 9 degrees of freedom.
Problem with your image
Long story short, I think the core issue with your image is that it fails to specify either a sform_code or a qform_code, and the two spatial representations are mutually exclusive:
fslhd ChickenEgg-PET-Data.nii
...
qform_name Unknown
qform_code 0
qto_xyz:1 0.865759 0.000000 0.000000 0.000000
qto_xyz:2 0.000000 0.865759 0.000000 0.000000
qto_xyz:3 0.000000 0.000000 0.796000 0.000000
qto_xyz:4 0.000000 0.000000 0.000000 1.000000
qform_xorient Left-to-Right
qform_yorient Posterior-to-Anterior
qform_zorient Inferior-to-Superior
sform_name Unknown
sform_code 2
sto_xyz:1 1.000000 0.000000 0.000000 -54.975697
sto_xyz:2 0.000000 1.000000 0.000000 -54.975697
sto_xyz:3 0.000000 0.000000 1.000000 -37.411999
sto_xyz:4 0.000000 0.000000 0.000000 1.000000
Solution
The easiest fix is would be to make sure the SForm and QForm store the same representation (e.g. the QForm looks plausible but adding a translation to set the origin is required) and set each to the QForm_code and SForm_code to scanner_anat
.
Longer term, it looks like your data was acquired on a CTI Concorde model 2501. I have never seen the text format for the header you provided. I would see if you can either export the data from the scanner as DICOM or ECAT. You can use dcm2niix to convert those to clean NIfTI or NRRD datasets. dcm2niix can be run from the command line, but it is also available as a slice extension. You might also want to look at PET2BIDS which is a wrapper from dcm2niix that helps specify a lot of the PET details that are useful for curated archival datasets.