Jonckheere-Terpstra interpretation I am running the Jonckheere-Terpstra in place of Kruskal-Wallis test, as my factor is in ordinal scale (i.e. groups can be ordered).
The Asymptotic significance (2-tailed) is 0.000, so it seems there is a trend in the response variable, according to the trend of the factor.
However, if I look at the medians, this trend is not clear; in fact I have 4 levels in my factor and, in ascending order, their medians are: 0.1387, 0.2814, 0.5882, 0.3492.
So, I ran a Mann Withney test with Bonferroni correction to check if the difference between last two pairs is significant and it is NOT. 
Two questions: 
1) Am I doing it right?
2) I ran the J-T test for another response variable, which is ordered conversely with respect to the factor (i.e. if I increase the factor, the response variable decreases). If the J-T statistic is NEGATIVE and the p-value is 0.000, can I conclude that the response variable is DECREASING (instead of increasing) according to a factor increase?
Thanks.
 A: First, one note. Like Kruskal-Wallis, Jonckheere-Terpstra is generally not a test of medians. It is a test of distributional "locations", or stochastic prevalence. It can give significant result even if medians are equal.
Now, for your test (let us honour your medians as locations). The test is highly significant because there is expressed trend in 0.1387, 0.2814, 0.5882. You tested the "all groups equal" null against the alternative hypothesis that, in population, Lev1<=Lev2<=Lev3<=Lev4 with at least one of the inequalities is strict (<). Jonckheere-Terpstra, unlike Kruskal-Wallis, considers only such a monotonic alternative hypothesis, not curvilinear one such as Lev1<=Lev2<=Lev3>=Lev4 for example. Therefore, under J-T pairwise comparisons, Lev3>Lev4 never will be significant. It is the constraint. Under K-W pairwise comparisons, it can be significant, certainly.
Please see this answer for particulars what hypotheses Jonckheere-Terpstra test tests and what are their p values.
A: The J-T test combines all the comparisons into a single test so it doesn't really make sense to extract a specific conclusion about one comparison. Specifically, the J-T test statistic is the sum of a set of two-sample rank-sum statistics. For your data, the reversal in trend between levels 3 and 4 will reduce the J-T statistic (the rank-sum statistic from this comparison will be negative), but clearly the overall positive trend outweighs this, giving a large J-T statistic, a small P-value, and favoring the alternative hypothesis. As ttnphns has written, the alternative hypothesis is a string of <= with at least one strict <. So your K-W result is not inconsistent with the alternative hypothesis of the J-T test. A pattern of true locations of 1 < 2 < 3 = 4 would be compatible with your medians, the J-T alternative hypothesis, and the K-W null hypothesis.
Of course another plausible true pattern could be 1 < 2 < 3 > 4, which is not consistent with either hypothesis, null or alternative, of the J-T test. One price for the added power of a more specific alternative is the possibility that it will miss the target, that the true pattern will not match either the null or the alternative.
But also if you had a legitimate a priori reason to hypothesize a peaked pattern of 1 < 2 < 3 > 4, there's a variant of the J-T test, due to Mack & Wolfe ("K-sample rank tests for umbrella alternatives" JASA 76:175-181, 1981), that tests for this.
