If we say "Survival difference adjusted for covariates" do we mean just additive adjustment, or allow also the interactions? Let's make it independent from any statistical package, so I will provide just numbers for the illustration. Maybe this give you some clues.
I want to compare two survival curves, for group A and B.
Unadjusted for anything, the HR taken from the Cox model (hazards are proportional quite well) is: 3.5
I have 2 covariates:

*

*Age (discretized <50 years and >= 50 years; Not my decision, please don't ask me to not discretize)

*Sex (Male and Female)

When entered in additive way (in R it would be: ~ AB + Sex + Gender), meaning no interaction between them, I get the HR for the AB = 3  (it decreases).
When I check only the Age + Sex (no AB here), I get that they both are statistically significant p<0.001 with high HR, like 3.5 - 4
There is no interaction between them showed by the model (A*B shows very small effect on the A:B term).
So, using just AB flag gives HR = 3.5
AB adjusted for Age + Sex gives HR = 3
But(!) AB crossed with Age and Sex, i.e. AB * Sex * Age gives HR = 6!
And now a few interactions are significant as well.
What's funny, the concordance is almost the same, so I assume the unexplained deviance remains similar. The covariates share some of the effect mutually.
Now, my question is:
When I am asked to "adjust for a covariate", do they mean:

*

*adjust in additive manner, assuming there is no interaction between them, they are "orthogonal", sharing additive parts (mutually exclusive) of the unexplained variance/deviance


*adjust in any manner that makes sense, including some interactions, if the model fit is best and it makes sense. For example I can really observe differences at some level of covariates (simple effects)?
If only additive adjustment is allowed, what about the ignored strong (I speak only about really big, visible ones) interactions?
In other words, which model is "adjusted for covariates":

*

*AB + Sex + Gender  (sex and gender are not of interest)


*AB + strata(Sex) + strata(Gender)


*AB * Sex * Gender


*AB + Sex * Gender?
 A: In principle, "adjusted for covariates" can mean any of the above. That has two major implications.
First, when you are reading (or reviewing) a paper, you need to make sure you understand just which "adjustment" the authors used. I suspect that it's usually an additive adjustment without interactions, but I can't claim to have data on relative prevalence of various "adjustment" types.
Second, when you are trying to "adjust for covariates" you need to make conscious choices. Additive (at least) terms are generally more informative than stratification, provided that you don't lose proportional hazards with the adjustment variable.
Think about adding interactions for adjustment just as you might think about adding other predictors, because that's what they are. Do the interactions make sense to include based on your understanding of the subject matter? For example, the NIH typically expect investigators to evaluate sex-specific effects, which requires an interaction term. Do you then risk overfitting by evaluating more predictors than are warranted by the number of events? If both answers are "yes," might you want to penalize the coefficients of "adjusted for" predictors as one might with ridge regression?
A few warnings, in no particular order:
(A) A change of hazard ratio (HR) from 3.5 to 3 might not really be a significant decrease. Examine the standard errors, not just the point estimates.
(B) Concordance and "unexplained deviance" measure different things. The former is just how well you predict the rank-order of event times; the latter is related to more global model fit. The Akaike Information Criterion (AIC) is generally available from model results and evaluates the latter corrected for the number of coefficient values estimated by the model.
(C) I'm assuming that when you wrote "Sex and Gender" in many places you actually meant "Sex and Age" as in your first example. There would be a good deal of collinearity between Sex and Gender, wreaking havoc with the precision of coefficient estimates (even if predictions from the model would end up OK). If you want to do such modeling, it might be better to model Sex as chromosomal sex and then include another predictor for transgender (etc) status, along with an interaction between them.
(D) If your adjustment variables are Age and Sex then you should have many events in all combinations of them with the A,B groups. In some situations, however, you could end up with few or no events for some predictor-value combinations, leading to potential failure to converge in the modeling or very wide confidence limits for the interactions.
