If someone hits my mech, which has heavy armor and agility, in close combat does my mech take any damage?

From the SEC Card:

AGILITY (optional) ’Mech only.
This unit reduces to 1 all damage received from opposing units by
close combat, charge, ram, and death from above attacks.

HEAVY ARMOR Subtract 1 from any damage scored against this unit by close or ranged combat attacks. Heavy Armor does not reduce pushing, critical miss, charge, ram, or death from above damage.

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No, you do not. You still take 1 damage etc from charges etc, but in CC, if somebody hits you for 1, your agility reduces whatever damage to 1, and the heavy armor absorbs that 1, so you take 0 damage.

The short answer?

No, your mech does not take any damage.

The true answer is: your choice. You get to decide which defense modifier kicks in first AKA whether armor kicks in before or after the agility.

Right now I see no reason to do it otherwise but I could see something in the future that would cause someone going the other way if they need to advance in their dial.