eco synthesis metal only
ECO is any change to the design after layout is completed, or even after the parts are back, and is usually refers to a hand instantiated change. A metal change is an ECO that only affects interconnect, limiting the number of layers that have to change (and the cost). IPO is in-place-optimization, which is essentially upsizing/downsizing gates prior to tapeout, and is usually done semi-automatically by the PD tool, or physical synthesis tool.
IPO is least expensive since it's before tapeout.
Metal change can be next, since it limits the number of layers that change.
An ECO can be a metal change, or a full layer change, which is high in mask cost, and since it's usually hand instantiated, is expensive in engineering time. The advantage is that you dont disturb all of the other stuff that may be working.