Great question! It doesn't take on a different name after a downtrend (it's still a gravestone doji), but when it occurs after a strong downtrend it usually reflects capitulation because it can be a sign of renewed selling at any sign of strength. So if you see a gravestone doji after a strong downtrend, it's usually trapped longs trying to get out at any sign of strength to cut their losses.

At a market top, a gravestone doji is also just the higher timeframe version of a tweezer top. For example, a tweezer top on the 30-minute chart is essentially a gravestone doji on the 1-hour chart.