Lets say I have a design that is ~ drugTreatment + environmentalCondition + drugTreatment:environmentalCondition

with factors with the following levels:

drugTreatment:

1. control
2. A
3. B

environmentalCondition:

1. standard
2. heat
3. osmotic

For the interaction term drugTreatmentA:environmentalConditionheat, one could it interpret its meaning as:

The difference in effect of that drugTreatmentA has when the environment is environmentalConditionheat compared to drugTreatmentAs main effect (on base level environment).

BUT:

I could also interpret the same interaction term as:

The difference in effect of that environmentalConditionheat has when the drug is drugTreatmentA compared to environmentalConditionheats main effect (base level drug).

Is this line of reasoning correct? Intuitivly I feel as if these values could be different.

In terms of design, I guess my question is:

for Design 1: ~ drugTreatment + environmentalCondition + drugTreatment:environmentalCondition

and Design 2: ~ environmentalCondition + drugTreatment + environmentalCondition:drugTreatment

are the interaction terms drugTreatmentA:environmentalConditionheat and environmentalConditionheat:drugTreatmentA the same?

If so, can someone please help me understand why this is the case?

In example 3 of the man page of ?results it makes reference to "the main effect", in my case, I don't know if the environmnet, or the drug is the main effect.

To make my question clearer, I created a diagram, I'd like to understand why the pink and red arrows are the same.

https://imgur.com/a/XshRM

I think I may have graphically answered my own question.

The interaction term in the design stated in my question is actually testing something different from what I originally thought. See new updated image below.

https://imgur.com/8twNdmn

If you wanted to test the coeficients I was talking about before, one would use ~environmentalCondition + environmentalCondition:drugTreatment and ~drugTreatment + drugTreatment:environmentalCondition