The principle of subsidiarity is restated from earlier documents in the Catechism:
1883. ... The teaching of the Church has elaborated the principle of subsidiarity, according to which "a community of a higher order should not interfere in the internal life of a community of a lower order, depriving the latter of its functions, but rather should support it in case of need and help to co-ordinate its activity with the activities of the rest of society, always with a view to the common good."7
7 Centesimus annus 48 §4; cf. Pius X Quadragesimo anno I, 184–186
Reciprocal subsidiarity is — on the face of it — meaningless. While subsidiarity is asserting the independence of a lower order against the interference of a higher order, that higher order is immune to the lower order's interference simply by virtue of being the higher order.
But subsidiarity is not just protection from interference. It is the support and co-ordination of activities with the rest of society, always with a view to the common good. Subsidiarity commonly means that the government should guide lower communities to work together to the common good.
It is this which can be reciprocal. Governments should listen to their communities to understand what the common good actually is. Building a nation is not pure imposition (which directly goes against subsidiarity) or guiding/defining boundaries which allow local determination but which does not further the common good, pitting local governments against each other and the State*. The common good is the good of all. Everyone has to put themselves at the service of others, within their own sphere of responsibility.
The Pope was addressing both Houses of the Legislature in what is arguably the most powerful country in the world. He was saying that for a government to be a good government, it has to allow itself to be guided by its people in order that it can do as it must and serve the good of the people.
*State here means the top level of government: Federal in the US, Parliament in the UK, etc.
