Drag and drop items in RecyclerView with GridLayoutManager

This is my solution with database reordering:

    ItemTouchHelper.SimpleCallback simpleItemTouchCallback = new ItemTouchHelper.SimpleCallback(ItemTouchHelper.UP | ItemTouchHelper.DOWN, ItemTouchHelper.LEFT | ItemTouchHelper.RIGHT) {
        @Override
        public boolean onMove(RecyclerView recyclerView, RecyclerView.ViewHolder viewHolder, RecyclerView.ViewHolder target) {
            final int fromPosition = viewHolder.getAdapterPosition();
            final int toPosition = target.getAdapterPosition();
            if (fromPosition < toPosition) {
                for (int i = fromPosition; i < toPosition; i++) {
                    Collections.swap(mAdapter.getCapitolos(), i, i + 1);
                }
            } else {
                for (int i = fromPosition; i > toPosition; i--) {
                    Collections.swap(mAdapter.getCapitolos(), i, i - 1);
                }
            }
            mAdapter.notifyItemMoved(fromPosition, toPosition);
            return true;
        }

        @Override
        public void onSwiped(RecyclerView.ViewHolder viewHolder, int swipeDir) {
            MyViewHolder svH = (MyViewHolder ) viewHolder;
            int index = mAdapter.getCapitolos().indexOf(svH.currentItem);
            mAdapter.getCapitolos().remove(svH.currentItem);
            mAdapter.notifyItemRemoved(index);
            if (emptyView != null) {
                if (mAdapter.getCapitolos().size() > 0) {
                emptyView.setVisibility(TextView.GONE);
                } else {
                emptyView.setVisibility(TextView.VISIBLE);
                }
            }
        }

        @Override
        public void clearView(RecyclerView recyclerView, RecyclerView.ViewHolder viewHolder) {
            super.clearView(recyclerView, viewHolder);
            reorderData();
        }
    };

    ItemTouchHelper itemTouchHelper = new ItemTouchHelper(simpleItemTouchCallback);
    itemTouchHelper.attachToRecyclerView(recList);

There is a support functions tahat make use of AsyncTask:

private void reorderData() {
    AsyncTask<String, Void, Spanned> task = new AsyncTask<String, Void, Spanned>() {
        @Override
        protected Spanned doInBackground(String... strings) {
            dbService.deleteAllData();
            for (int i = mAdapter.getCapitolos().size() - 1; i >= 0; i--) {
                Segnalibro s = mAdapter.getCapitolos().get(i);
                dbService.saveData(s.getIdCapitolo(), s.getVersetto());
            }
            return null;
        }

        @Override
        protected void onPostExecute(Spanned spanned) {
        }
    };
    task.execute();
}

There is actually a better way to achieve this. You can use some of the RecyclerView's "companion" classes:

ItemTouchHelper, which is

a utility class to add swipe to dismiss and drag & drop support to RecyclerView.

and its ItemTouchHelper.Callback, which is

the contract between ItemTouchHelper and your application

// Create an `ItemTouchHelper` and attach it to the `RecyclerView`
ItemTouchHelper ith = new ItemTouchHelper(_ithCallback);
ith.attachToRecyclerView(rv);

// Extend the Callback class
ItemTouchHelper.Callback _ithCallback = new ItemTouchHelper.Callback() {
    //and in your imlpementaion of
    public boolean onMove(RecyclerView recyclerView, RecyclerView.ViewHolder viewHolder, RecyclerView.ViewHolder target) {
        // get the viewHolder's and target's positions in your adapter data, swap them
        Collections.swap(/*RecyclerView.Adapter's data collection*/, viewHolder.getAdapterPosition(), target.getAdapterPosition());
        // and notify the adapter that its dataset has changed
        _adapter.notifyItemMoved(viewHolder.getAdapterPosition(), target.getAdapterPosition());
        return true;
    }

    @Override
    public void onSwiped(RecyclerView.ViewHolder viewHolder, int direction) {
        //TODO    
    }

    //defines the enabled move directions in each state (idle, swiping, dragging). 
    @Override
    public int getMovementFlags(RecyclerView recyclerView, RecyclerView.ViewHolder viewHolder) {
        return makeFlag(ItemTouchHelper.ACTION_STATE_DRAG,
                ItemTouchHelper.DOWN | ItemTouchHelper.UP | ItemTouchHelper.START | ItemTouchHelper.END);
    }
};

For more details check their documentation.