The order is not a tradition; it is in the regulatory text, and there is a physical reason for it.
The sequence. When testing for atmospheric hazards, test first for oxygen, then for combustible gases and vapors, and then for toxic gases and vapors (29 CFR 1910.146(d)(5)(iii)). Most combustible-gas sensors need adequate oxygen to read correctly, and some toxic-gas readings are affected by both, so the sequence is what makes the later numbers trustworthy.
Before and during. The employer's permit program must test conditions in the space to determine whether acceptable entry conditions exist before entry is authorized, with a narrow allowance for spaces that cannot be isolated, such as sewers, where pre-entry testing is done to the extent feasible (29 CFR 1910.146(d)(5)(i)). The space must then be tested or monitored as necessary to determine that acceptable conditions are being maintained throughout the entry (29 CFR 1910.146(d)(5)(ii)).
The entrant's rights. Each authorized entrant, or that employee's authorized representative, must be given an opportunity to observe the pre-entry and any subsequent testing (29 CFR 1910.146(d)(5)(iv)), may request a reevaluation if they have reason to believe the evaluation was inadequate, which must then be done in their presence ((d)(5)(v)), and must be provided the results immediately on request ((d)(5)(vi)). The results, with the tester's initials and times, go on the permit (29 CFR 1910.146(f)(10)).
The numbers. A hazardous atmosphere includes oxygen below 19.5 percent or above 23.5 percent (29 CFR 1910.146(b)), a flammable gas, vapor or mist above 10 percent of its lower flammable limit, or a substance above its permissible exposure limit.
Entrants who have never been offered the chance to watch the meter are the quickest way to find out a program is paperwork.